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RD Sharma solutions for Class 12 Mathematics chapter 22 - Differential Equations

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

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RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

Chapter 22: Differential Equations

Chapter 22: Differential Equations solutions [Pages 4 - 5]

Q 1 | Page 4
\[\frac{d^3 x}{d t^3} + \frac{d^2 x}{d t^2} + \left( \frac{dx}{dt} \right)^2 = e^t\]
Q 2 | Page 4
\[\frac{d^2 y}{d x^2} + 4y = 0\]
Q 3 | Page 5
\[\left( \frac{dy}{dx} \right)^2 + \frac{1}{dy/dx} = 2\]
Q 4 | Page 5
\[\sqrt{1 + \left( \frac{dy}{dx} \right)^2} = \left( c\frac{d^2 y}{d x^2} \right)^{1/3}\]
Q 5 | Page 5
\[\frac{d^2 y}{d x^2} + \left( \frac{dy}{dx} \right)^2 + xy = 0\]
Q 6 | Page 5
\[\sqrt[3]{\frac{d^2 y}{d x^2}} = \sqrt{\frac{dy}{dx}}\]
Q 7 | Page 5
\[\frac{d^4 y}{d x^4} = \left\{ c + \left( \frac{dy}{dx} \right)^2 \right\}^{3/2}\]
Q 8 | Page 5
\[x + \left( \frac{dy}{dx} \right) = \sqrt{1 + \left( \frac{dy}{dx} \right)^2}\]
Q 9 | Page 5
\[y\frac{d^2 x}{d y^2} = y^2 + 1\]
Q 10 | Page 5
\[s^2 \frac{d^2 t}{d s^2} + st\frac{dt}{ds} = s\]
Q 11 | Page 5
\[x^2 \left( \frac{d^2 y}{d x^2} \right)^3 + y \left( \frac{dy}{dx} \right)^4 + y^4 = 0\]
Q 12 | Page 5
\[\frac{d^3 y}{d x^3} + \left( \frac{d^2 y}{d x^2} \right)^3 + \frac{dy}{dx} + 4y = \sin x\]
Q 13 | Page 5

(xy2 + x) dx + (y − x2y) dy = 0

Q 14 | Page 5
\[\sqrt{1 - y^2} dx + \sqrt{1 - x^2} dx = 0\]
Q 15 | Page 5
\[\frac{d^2 y}{d x^2} = \left( \frac{dy}{dx} \right)^{2/3}\]
Q 16 | Page 5
\[2\frac{d^2 y}{d x^2} + 3\sqrt{1 - \left( \frac{dy}{dx} \right)^2 - y} = 0\]
Q 17 | Page 5
\[5\frac{d^2 y}{d x^2} = \left\{ 1 + \left( \frac{dy}{dx} \right)^2 \right\}^{3/2}\]
Q 18 | Page 5
\[y = x\frac{dy}{dx} + a\sqrt{1 + \left( \frac{dy}{dx} \right)^2}\]
Q 19 | Page 5
\[y = px + \sqrt{a^2 p^2 + b^2},\text{ where p} = \frac{dy}{dx}\]
Q 20 | Page 5
\[\frac{d^2 y}{d x^2} + 3 \left( \frac{dy}{dx} \right)^2 = x^2 \log\left( \frac{d^2 y}{d x^2} \right)\]
Q 21 | Page 5
\[\left( \frac{d^2 y}{d x^2} \right)^2 + \left( \frac{dy}{dx} \right)^2 = x \sin \left( \frac{d^2 y}{d x^2} \right)\]
Q 22 | Page 5

(y'')2 + (y')3 + sin y = 0

Q 23 | Page 5
\[\frac{d^2 y}{d x^2} + 5x\left( \frac{dy}{dx} \right) - 6y = \log x\]
Q 24 | Page 5
\[\frac{d^3 y}{d x^3} + \frac{d^2 y}{d x^2} + \frac{dy}{dx} + y \sin y = 0\]
Q 25 | Page 5
\[\frac{dy}{dx} + e^y = 0\]
Q 26 | Page 5
\[\left( \frac{dy}{dx} \right)^3 - 4 \left( \frac{dy}{dx} \right)^2 + 7y = \sin x\]

Chapter 22: Differential Equations solutions [Pages 16 - 17]

Q 1 | Page 16

Form the differential equation of the family of curves represented by y2 = (x − c)3.

Q 2 | Page 16

Form the differential equation corresponding to y = emx by eliminating m.

Q 3.1 | Page 16

Form the differential equation from the following primitive where constants are arbitrary:
y2 = 4ax

Q 3.2 | Page 16

Form the differential equation from the following primitive where constants are arbitrary:
y = cx + 2c2 + c3

Q 3.3 | Page 16

Form the differential equation from the following primitive where constants are arbitrary:
xy = a2

Q 3.4 | Page 16

Form the differential equation from the following primitive where constants are arbitrary:
y = ax2 + bx + c

Q 4 | Page 16

Find the differential equation of the family of curves y = Ae2x + Be−2x, where A and B are arbitrary constants.

Q 5 | Page 16

Find the differential equation of the family of curves, x = A cos nt + B sin nt, where A and B are arbitrary constants.

Q 6 | Page 16

Form the differential equation corresponding to y2 = a (b − x2) by eliminating a and b.

Q 7 | Page 16

Form the differential equation corresponding to y2 − 2ay + x2 = a2 by eliminating a.

Q 8 | Page 16

Form the differential equation corresponding to (x − a)2 + (y − b)2 = r2 by eliminating a and b.

Q 9 | Page 17

Find the differential equation of all the circles which pass through the origin and whose centres lie on y-axis.

Q 10 | Page 17

Find the differential equation of all the circles which pass through the origin and whose centres lie on x-axis.

Q 11 | Page 17

Assume that a rain drop evaporates at a rate proportional to its surface area. Form a differential equation involving the rate of change of the radius of the rain drop.

 
Q 12 | Page 17

Find the differential equation of all the parabolas with latus rectum '4a' and whose axes are parallel to x-axis.

Q 13 | Page 17

Show that the differential equation of which y = 2(x2 − 1) + \[c e^{- x^2}\] is a solution, is \[\frac{dy}{dx} + 2xy = 4 x^3\]

Q 14 | Page 17

Form the differential equation having \[y = \left( \sin^{- 1} x \right)^2 + A \cos^{- 1} x + B\], where A and B are arbitrary constants, as its general solution.

Q 15.1 | Page 17

Form the differential equation of the family of curves represented by the equation (a being the parameter):
(2x + a)2 + y2 = a2

Q 15.2 | Page 17

Form the differential equation of the family of curves represented by the equation (a being the parameter):
(2x − a)2 − y2 = a2

Q 15.3 | Page 17

Form the differential equation of the family of curves represented by the equation (a being the parameter):
 (x − a)2 + 2y2 = a2

Q 16.01 | Page 17

Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
x2 + y2 = a2

Q 16.02 | Page 17

Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
x2 − y2 = a2

Q 16.03 | Page 17

Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
y2 = 4ax

Q 16.04 | Page 17

Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
x2 + (y − b)2 = 1

Q 16.05 | Page 17

Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
(x − a)2 − y2 = 1

Q 16.06 | Page 17

Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):

\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\]

 

Q 16.07 | Page 17

Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
y2 = 4a (x − b)

 

Q 16.08 | Page 17

Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
y = ax3

Q 16.09 | Page 17

Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
x2 + y2 = ax3

Q 16.1 | Page 17

Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
y = eax

Q 17 | Page 17

Form the differential equation representing the family of ellipses having centre at the origin and foci on x-axis.

Q 18 | Page 17

Form the differential equation of the family of hyperbolas having foci on x-axis and centre at the origin.

Q 19 | Page 17

Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.

Chapter 22: Differential Equations solutions [Pages 24 - 25]

Q 1 | Page 24

Show that y = bex + ce2x is a solution of the differential equation,
\[\frac{d^2 y}{d x^2} - 3\frac{dy}{dx} + 2y = 0\]

Q 2 | Page 24

Verify that y = 4 sin 3x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + 9y = 0\]

Q 3 | Page 24

Show that the function y = A cos x + B sin x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + y = 0\]

Q 4 | Page 24

Show that the function y = A cos x + B sin x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + y = 0\]

Q 5 | Page 25

Show that the function y = A cos 2x − B sin 2x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + 4y = 0\].

Q 6 | Page 25

Show that y = AeBx is a solution of the differential equation

\[\frac{d^2 y}{d x^2} = \frac{1}{y} \left( \frac{dy}{dx} \right)^2\]
Q 7 | Page 25

Verify that y = \[\frac{a}{x} + b\] is a solution of the differential equation
\[\frac{d^2 y}{d x^2} + \frac{2}{x}\left( \frac{dy}{dx} \right) = 0\]

Q 8 | Page 25

Verify that y2 = 4ax is a solution of the differential equation y = x \[\frac{dy}{dx} + a\frac{dx}{dy}\]

Q 9 | Page 25

Show that Ax2 + By2 = 1 is a solution of the differential equation x \[\left\{ y\frac{d^2 y}{d x^2} + \left( \frac{dy}{dx} \right)^2 \right\} = y\frac{dy}{dx}\]

 

Q 10 | Page 25

Show that y = ax3 + bx2 + c is a solution of the differential equation \[\frac{d^3 y}{d x^3} = 6a\].

 

Q 11 | Page 25

Hence, the given function is the solution to the given differential equation. \[\frac{c - x}{1 + cx}\] is a solution of the differential equation (1 + x2) \[\frac{dy}{dx}\].

Q 12 | Page 25

Show that y = ex (A cos x + B sin x) is the solution of the differential equation \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + 2y = 0\]

Q 13 | Page 25

Verify that y = cx + 2c2 is a solution of the differential equation 

\[2 \left( \frac{dy}{dx} \right)^2 + x\frac{dy}{dx} - y = 0\].
Q 14 | Page 25

Verify that y = − x − 1 is a solution of the differential equation (y − x) dy − (y2 − x2) dx = 0.

Q 15 | Page 25

Verify that y2 = 4a (x + a) is a solution of the differential equations
\[y\left\{ 1 - \left( \frac{dy}{dx} \right)^2 \right\} = 2x\frac{dy}{dx}\]

Q 16 | Page 25

Verify that \[y = ce^{tan^{- 1}} x\]  is a solution of the differential equation \[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + \left( 2x - 1 \right)\frac{dy}{dx} = 0\]

Q 17 | Page 25

Verify that \[y = e^{m \cos^{- 1} x}\] satisfies the differential equation \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} - m^2 y = 0\]

Q 18 | Page 25

Verify that y = log \[\left( x + \sqrt{x^2 + a^2} \right)^2\]  satisfies the differential equation \[\left( a^2 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = 0\]

Q 19 | Page 25

Show that the differential equation of which \[y = 2\left( x^2 - 1 \right) + c e^{- x^2}\]  is a solution is \[\frac{dy}{dx} + 2xy = 4 x^3\]

Q 20 | Page 25

Show that y = e−x + ax + b is solution of the differential equation\[e^x \frac{d^2 y}{d x^2} = 1\]

 

Q 21.1 | Page 25

For the following differential equation verify that the accompanying function is a solution:

Differential equation Function
\[x\frac{dy}{dx} = y\]
y = ax
Q 21.2 | Page 25

For the following differential equation verify that the accompanying function is a solution:

Differential equation Function
\[x + y\frac{dy}{dx} = 0\]
\[y = \pm \sqrt{a^2 - x^2}\]
Q 21.3 | Page 25

For the following differential equation verify that the accompanying function is a solution:

Differential equation Function
\[x\frac{dy}{dx} + y = y^2\]
\[y = \frac{a}{x + a}\]
Q 21.4 | Page 25

For the following differential equation verify that the accompanying function is a solution:

Differential equation Function
\[x^3 \frac{d^2 y}{d x^2} = 1\]
\[y = ax + b + \frac{1}{2x}\]
Q 21.5 | Page 25

For the following differential equation verify that the accompanying function is a solution:

Differential equation Function
\[y = \left( \frac{dy}{dx} \right)^2\]
\[y = \frac{1}{4} \left( x \pm a \right)^2\]

Chapter 22: Differential Equations solutions [Page 28]

Q 1 | Page 28

Differential equation \[x\frac{dy}{dx} = 1, y\left( 1 \right) = 0\]

Function y = log x

Q 2 | Page 28

Differential equation \[\frac{dy}{dx} = y, y\left( 0 \right) = 1\]
Function y = ex

Q 3 | Page 28

Differential equation \[\frac{d^2 y}{d x^2} + y = 0, y \left( 0 \right) = 0, y' \left( 0 \right) = 1\] Function y = sin x

Q 4 | Page 28

Differential equation \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} = 0, y \left( 0 \right) = 2, y'\left( 0 \right) = 1\] Function y = ex + 1

Q 5 | Page 28

Differential equation \[\frac{dy}{dx} + y = 2, y \left( 0 \right) = 3\] Function y = e−x + 2

Q 6 | Page 28

Differential equation \[\frac{d^2 y}{d x^2} + y = 0, y \left( 0 \right) = 1, y' \left( 0 \right) = 1\] Function y = sin x + cos x

Q 7 | Page 28

Differential equation \[\frac{d^2 y}{d x^2} - y = 0, y \left( 0 \right) = 2, y' \left( 0 \right) = 0\] Function y = ex + ex

Q 8 | Page 28

Differential equation \[\frac{d^2 y}{d x^2} - 3\frac{dy}{dx} + 2y = 0, y \left( 0 \right) = 1, y' \left( 0 \right) = 3\] Function y = ex + e2x

Q 9 | Page 28

Differential equation \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + y = 0, y \left( 0 \right) = 1, y' \left( 0 \right) = 2\] Function y = xex + ex

Chapter 22: Differential Equations solutions [Page 34]

Q 1 | Page 34
\[\frac{dy}{dx} = x^2 + x - \frac{1}{x}, x \neq 0\]
Q 2 | Page 34
\[\frac{dy}{dx} = x^5 + x^2 - \frac{2}{x}, x \neq 0\]
Q 3 | Page 34
\[\frac{dy}{dx} + 2x = e^{3x}\]
Q 4 | Page 34
\[\left( x^2 + 1 \right)\frac{dy}{dx} = 1\]
Q 5 | Page 34
\[\frac{dy}{dx} = \frac{1 - \cos x}{1 + \cos x}\]
Q 6 | Page 34
\[\left( x + 2 \right)\frac{dy}{dx} = x^2 + 3x + 7\]
Q 7 | Page 34
\[\frac{dy}{dx} = \tan^{- 1} x\]

Q 8 | Page 34
\[\frac{dy}{dx} = \log x\]

Q 9 | Page 34
\[\frac{1}{x}\frac{dy}{dx} = \tan^{- 1} x, x \neq 0\]
Q 10 | Page 34
\[\frac{dy}{dx} = \cos^3 x \sin^2 x + x\sqrt{2x + 1}\]
Q 11 | Page 34

(sin x + cos x) dy + (cos x − sin x) dx = 0

Q 12 | Page 34
\[\frac{dy}{dx} - x \sin^2 x = \frac{1}{x \log x}\]
Q 13 | Page 34
\[\frac{dy}{dx} = x^5 \tan^{- 1} \left( x^3 \right)\]
Q 14 | Page 34
\[\sin^4 x\frac{dy}{dx} = \cos x\]
Q 15 | Page 34
\[\cos x\frac{dy}{dx} - \cos 2x = \cos 3x\]
Q 16 | Page 34
\[\sqrt{1 - x^4} dy = x dx\]
Q 17 | Page 34
\[\sqrt{a + x} dy + x dx = 0\]
Q 18 | Page 34
\[\left( 1 + x^2 \right)\frac{dy}{dx} - x = 2 \tan^{- 1} x\]
Q 19 | Page 34
\[\frac{dy}{dx} = x \log x\]
Q 20 | Page 34
\[\frac{dy}{dx} = x e^x - \frac{5}{2} + \cos^2 x\]
Q 21 | Page 34
\[\left( x^3 + x^2 + x + 1 \right)\frac{dy}{dx} = 2 x^2 + x\]
Q 22 | Page 34
\[\sin\left( \frac{dy}{dx} \right) = k ; y\left( 0 \right) = 1\]
Q 23 | Page 34
\[e^{dy/dx} = x + 1 ; y\left( 0 \right) = 3\]
Q 24 | Page 34

C' (x) = 2 + 0.15 x ; C(0) = 100

Q 25 | Page 34
\[x\frac{dy}{dx} + 1 = 0 ; y \left( - 1 \right) = 0\]
Q 26 | Page 34
\[x\left( x^2 - 1 \right)\frac{dy}{dx} = 1, y\left( 2 \right) = 0\]

Chapter 22: Differential Equations solutions [Page 38]

Q 1 | Page 38
\[\frac{dy}{dx} + \frac{1 + y^2}{y} = 0\]
Q 2 | Page 38
\[\frac{dy}{dx} = \frac{1 + y^2}{y^3}\]
Q 3 | Page 38
\[\frac{dy}{dx} = \sin^2 y\]
Q 4 | Page 38
\[\frac{dy}{dx} = \frac{1 - \cos 2y}{1 + \cos 2y}\]

Chapter 22: Differential Equations solutions [Pages 55 - 57]

Q 1 | Page 55
\[\left( x - 1 \right)\frac{dy}{dx} = 2 xy\]
Q 2 | Page 55

(1 + x2) dy = xy dx

Q 3 | Page 55
\[\frac{dy}{dx} = \left( e^x + 1 \right) y\]
Q 4 | Page 55
\[\left( x - 1 \right)\frac{dy}{dx} = 2 x^3 y\]
Q 5 | Page 55

xy (y + 1) dy = (x2 + 1) dx

Q 6 | Page 55
\[5\frac{dy}{dx} = e^x y^4\]
Q 7 | Page 55

x cos y dy = (xex log x + ex) dx

Q 8 | Page 55
\[\frac{dy}{dx} = e^{x + y} + x^2 e^y\]
Q 9 | Page 55
\[x\frac{dy}{dx} + y = y^2\]
Q 10 | Page 55

(ey + 1) cos x dx + ey sin x dy = 0

Q 11 | Page 55

x cos2 y  dx = y cos2 x dy

Q 12 | Page 55

xy dy = (y − 1) (x + 1) dx

Q 13 | Page 55
\[x\frac{dy}{dx} + \cot y = 0\]
Q 14 | Page 55
\[\frac{dy}{dx} = \frac{x e^x \log x + e^x}{x \cos y}\]
Q 15 | Page 55
\[\frac{dy}{dx} = e^{x + y} + e^y x^3\]
Q 16 | Page 55
\[y\sqrt{1 + x^2} + x\sqrt{1 + y^2}\frac{dy}{dx} = 0\]
Q 17 | Page 55
\[\sqrt{1 + x^2} dy + \sqrt{1 + y^2} dx = 0\]
Q 18 | Page 55
\[\sqrt{1 + x^2 + y^2 + x^2 y^2} + xy\frac{dy}{dx} = 0\]
Q 19 | Page 55
\[\frac{dy}{dx} = \frac{e^x \left( \sin^2 x + \sin 2x \right)}{y\left( 2 \log y + 1 \right)}\]
Q 20 | Page 55
\[\frac{dy}{dx} = \frac{x\left( 2 \log x + 1 \right)}{\sin y + y \cos y}\]
Q 21 | Page 55

(1 − x2) dy + xy dx = xy2 dx

Q 22 | Page 55

tan y dx + sec2 y tan x dy = 0

Q 23 | Page 55

(1 + x) (1 + y2) dx + (1 + y) (1 + x2) dy = 0

Q 24 | Page 55

tan y \[\frac{dy}{dx}\] = sin (x + y) + sin (x − y) 

 

Q 25 | Page 55
\[\cos x \cos y\frac{dy}{dx} = - \sin x \sin y\]
Q 26 | Page 55
\[\frac{dy}{dx} + \frac{\cos x \sin y}{\cos y} = 0\]
Q 27 | Page 55
\[x\sqrt{1 - y^2} dx + y\sqrt{1 - x^2} dy = 0\]
Q 28 | Page 55

y (1 + ex) dy = (y + 1) ex dx

Q 29 | Page 55

(y + xy) dx + (x − xy2) dy = 0

Q 30 | Page 55
\[\frac{dy}{dx} = 1 - x + y - xy\]
Q 31 | Page 55

(y2 + 1) dx − (x2 + 1) dy = 0

Q 32 | Page 55

dy + (x + 1) (y + 1) dx = 0

Q 33 | Page 55
\[\frac{dy}{dx} = \left( 1 + x^2 \right)\left( 1 + y^2 \right)\]
Q 34 | Page 55
\[\left( x - 1 \right)\frac{dy}{dx} = 2 x^3 y\]
Q 35 | Page 55
\[\frac{dy}{dx} = e^{x + y} + e^{- x + y}\]
Q 36 | Page 55
\[\frac{dy}{dx} = \left( \cos^2 x - \sin^2 x \right) \cos^2 y\]
Q 37.1 | Page 55

Solve the following differential equation: 
(xy2 + 2x) dx + (x2 y + 2y) dy = 0

Q 37.2 | Page 55

Solve the following differential equation:
\[\text{ cosec }x \log y \frac{dy}{dx} + x^2 y^2 = 0\]

Q 38.1 | Page 55

Solve the following differential equation:
\[xy\frac{dy}{dx} = 1 + x + y + xy\]

 

Q 38.2 | Page 55

Solve the following differential equation:
\[y\left( 1 - x^2 \right)\frac{dy}{dx} = x\left( 1 + y^2 \right)\]

 

Q 38.3 | Page 55

Solve the following differential equation:
\[y e^\frac{x}{y} dx = \left( x e^\frac{x}{y} + y^2 \right)dy, y \neq 0\]

 

Q 38.4 | Page 55

Solve the following differential equation:
\[\left( 1 + y^2 \right) \tan^{- 1} xdx + 2y\left( 1 + x^2 \right)dy = 0\]

Q 39 | Page 56
\[\frac{dy}{dx} = y \tan 2x, y\left( 0 \right) = 2\] 
Q 40 | Page 56
\[2x\frac{dy}{dx} = 3y, y\left( 1 \right) = 2\]
Q 41 | Page 56
\[xy\frac{dy}{dx} = y + 2, y\left( 2 \right) = 0\]
Q 42 | Page 56
\[\frac{dy}{dx} = 2 e^x y^3 , y\left( 0 \right) = \frac{1}{2}\]
Q 43 | Page 56
\[\frac{dr}{dt} = - rt, r\left( 0 \right) = r_0\]
Q 44 | Page 56
\[\frac{dy}{dx} = y \sin 2x, y\left( 0 \right) = 1\]
Q 45.1 | Page 56
\[\frac{dy}{dx} = y \tan x, y\left( 0 \right) = 1\]
Q 45.2 | Page 56
\[2x\frac{dy}{dx} = 5y, y\left( 1 \right) = 1\]
Q 45.3 | Page 56
\[\frac{dy}{dx} = 2 e^{2x} y^2 , y\left( 0 \right) = - 1\]
Q 45.4 | Page 56
\[\cos y\frac{dy}{dx} = e^x , y\left( 0 \right) = \frac{\pi}{2}\]
Q 45.5 | Page 56
\[\frac{dy}{dx} = 2xy, y\left( 0 \right) = 1\]
Q 45.6 | Page 56
\[\frac{dy}{dx} = 1 + x^2 + y^2 + x^2 y^2 , y\left( 0 \right) = 1\]
Q 45.7 | Page 56
\[xy\frac{dy}{dx} = \left( x + 2 \right)\left( y + 2 \right), y\left( 1 \right) = - 1\]
Q 45.8 | Page 56
\[\frac{dy}{dx} = 1 + x + y^2 + x y^2\]when y = 0, x = 0
Q 45.9 | Page 56
\[2\left( y + 3 \right) - xy\frac{dy}{dx} = 0\], y(1) = −2
Q 46 | Page 56

Solve the differential equation \[x\frac{dy}{dx} + \cot y = 0\] \[y = \frac{\pi}{4}\] \[\sqrt{2}\]

Q 47 | Page 56

Solve the differential equation \[\left( 1 + x^2 \right)\frac{dy}{dx} + \left( 1 + y^2 \right) = 0\], given that y = 1, when x = 0.

Q 48 | Page 56

Solve the differential equation \[\frac{dy}{dx} = \frac{2x\left( \log x + 1 \right)}{\sin y + y \cos y}\], given that y = 0, when x = 1.

Q 49 | Page 56

Find the particular solution of edy/dx = x + 1, given that y = 3, when x = 0.

Q 50 | Page 56

Find the solution of the differential equation cos y dy + cos x sin y dx = 0 given that y = \[\frac{\pi}{2}\], when x = \[\frac{\pi}{2}\] 

 

Q 51 | Page 56

Find the particular solution of the differential equation \[\frac{dy}{dx} = - 4x y^2\]  given that y = 1, when x = 0.

Q 52 | Page 56

Find the equation of a curve passing through the point (0, 0) and whose differential equation is \[\frac{dy}{dx} = e^x \sin x\]

Q 53 | Page 56

For the differential equation xy \[\frac{dy}{dx}\] = (x + 2) (y + 2). Find the solution curve passing through the point (1, −1).

Q 54 | Page 56

The volume of a spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of the balloon after tseconds.

Q 55 | Page 56

In a bank principal increases at the rate of r% per year. Find the value of r if ₹100 double itself in 10 years (loge 2 = 0.6931).

Q 56 | Page 56

In a bank principal increases at the rate of 5% per year. An amount of Rs 1000 is deposited with this bank, how much will it worth after 10 years (e0.5 = 1.648).

Q 57 | Page 57

In a culture the bacteria count is 100000. The number is increased by 10% in 2 hours. In how many hours will the count reach 200000, if the rate of growth of bacteria is proportional to the number present.

Q 58 | Page 57

If y(x) is a solution of the different equation \[\left( \frac{2 + \sin x}{1 + y} \right)\frac{dy}{dx} = - \cos x\] and y(0) = 1, then find the value of y(π/2).

Q 59 | Page 57

Find the particular solution of the differential equation
(1 – y2) (1 + log x) dx + 2xy dy = 0, given that y = 0 when x = 1.

Chapter 22: Differential Equations solutions [Page 66]

Q 1 | Page 66
\[\frac{dy}{dx} = \left( x + y + 1 \right)^2\]
Q 2 | Page 66
\[\frac{dy}{dx}\cos\left( x - y \right) = 1\]
Q 3 | Page 66
\[\frac{dy}{dx} = \frac{\left( x - y \right) + 3}{2\left( x - y \right) + 5}\]
Q 4 | Page 66
\[\frac{dy}{dx} = \left( x + y \right)^2\]
Q 5 | Page 66
\[\left( x + y \right)^2 \frac{dy}{dx} = 1\]
Q 6 | Page 66
\[\cos^2 \left( x - 2y \right) = 1 - 2\frac{dy}{dx}\]
Q 7 | Page 66
\[\frac{dy}{dx} = \sec\left( x + y \right)\]
Q 8 | Page 66
\[\frac{dy}{dx} = \tan\left( x + y \right)\]
Q 9 | Page 66

(x + y) (dx − dy) = dx + dy

Q 10 | Page 66
\[\left( x + y + 1 \right)\frac{dy}{dx} = 1\]
Q 11 | Page 66
\[\frac{dy}{dx} + 1 = e^{x + y}\]

Chapter 22: Differential Equations solutions [Pages 83 - 84]

Q 1 | Page 83

x2 dy + y (x + y) dx = 0

Q 2 | Page 83
\[\frac{dy}{dx} = \frac{y - x}{y + x}\]
Q 3 | Page 83
\[\frac{dy}{dx} = \frac{y^2 - x^2}{2xy}\]
Q 4 | Page 83
\[x\frac{dy}{dx} = x + y\]
Q 5 | Page 83

(x2 − y2) dx − 2xy dy = 0

Q 6 | Page 83
\[\frac{dy}{dx} = \frac{x + y}{x - y}\]
Q 7 | Page 83
\[2xy\frac{dy}{dx} = x^2 + y^2\]
Q 8 | Page 83
\[x^2 \frac{dy}{dx} = x^2 - 2 y^2 + xy\]
Q 9 | Page 83
\[xy\frac{dy}{dx} = x^2 - y^2\]
Q 10 | Page 83

y ex/y dx = (xex/y + y) dy

Q 11 | Page 83

\[x^2 \frac{dy}{dx} = x^2 + xy + y^2 \]

Q 12 | Page 83

(y2 − 2xy) dx = (x2 − 2xy) dy

Q 13 | Page 83

2xy dx + (x2 + 2y2) dy = 0

Q 14 | Page 83

3x2 dy = (3xy + y2) dx

Q 15 | Page 83
\[\frac{dy}{dx} = \frac{x}{2y + x}\]
Q 16 | Page 83

(x + 2y) dx − (2x − y) dy = 0

Q 17 | Page 83
\[\frac{dy}{dx} = \frac{y}{x} - \sqrt{\frac{y^2}{x^2} - 1}\]
Q 18 | Page 83

Solve the following differential equations:
\[\frac{dy}{dx} = \frac{y}{x}\left\{ \log y - \log x + 1 \right\}\]

Q 19 | Page 83

y2 dx + (x2 − xy + y2) dy = 0

 
Q 20 | Page 83

y2 dx + (x2 − xy + y2) dy = 0

 
Q 21 | Page 83
\[\left[ x\sqrt{x^2 + y^2} - y^2 \right] dx + xy dy = 0\]
Q 22 | Page 83
\[x\frac{dy}{dx} = y - x \cos^2 \left( \frac{y}{x} \right)\]
Q 23 | Page 83
\[\frac{y}{x}\cos\left( \frac{y}{x} \right) dx - \left\{ \frac{x}{y}\sin\left( \frac{y}{x} \right) + \cos\left( \frac{y}{x} \right) \right\} dy = 0\]
Q 24 | Page 83
\[xy \log\left( \frac{x}{y} \right) dx + \left\{ y^2 - x^2 \log\left( \frac{x}{y} \right) \right\} dy = 0\]
Q 25 | Page 83
\[\left( 1 + e^{x/y} \right) dx + e^{x/y} \left( 1 - \frac{x}{y} \right) dy = 0\]
Q 26 | Page 83
\[\left( x^2 + y^2 \right)\frac{dy}{dx} = 8 x^2 - 3xy + 2 y^2\]
Q 27 | Page 83

(x2 − 2xy) dy + (x2 − 3xy + 2y2) dx = 0

Q 28 | Page 83
\[x\frac{dy}{dx} = y - x \cos^2 \left( \frac{y}{x} \right)\]
Q 29 | Page 83
\[x\frac{dy}{dx} - y = 2\sqrt{y^2 - x^2}\]
Q 30 | Page 83
\[x \cos\left( \frac{y}{x} \right) \cdot \left( y dx + x dy \right) = y \sin\left( \frac{y}{x} \right) \cdot \left( x dy - y dx \right)\]
Q 31 | Page 83

(x2 + 3xy + y2) dx − x2 dy = 0

Q 32 | Page 83
\[\left( x - y \right)\frac{dy}{dx} = x + 2y\]
Q 33 | Page 84

(2x2 y + y3) dx + (xy2 − 3x3) dy = 0

Q 34 | Page 84
\[x\frac{dy}{dx} - y + x \sin\left( \frac{y}{x} \right) = 0\]
Q 35 | Page 84
\[y dx + \left\{ x \log\left( \frac{y}{x} \right) \right\} dy - 2x dy = 0\]
Q 36.1 | Page 84

Solve the following initial value problem:
 (x2 + y2) dx = 2xy dy, y (1) = 0

Q 36.2 | Page 84

Solve the following initial value problem:
\[x e^{y/x} - y + x\frac{dy}{dx} = 0, y\left( e \right) = 0\]

Q 36.3 | Page 84

Solve the following initial value problem:
\[\frac{dy}{dx} - \frac{y}{x} + cosec\frac{y}{x} = 0, y\left( 1 \right) = 0\]

Q 36.4 | Page 84

Solve the following initial value problem:
(xy − y2) dx − x2 dy = 0, y(1) = 1

Q 36.5 | Page 84

Solve the following initial value problem:
\[\frac{dy}{dx} = \frac{y\left( x + 2y \right)}{x\left( 2x + y \right)}, y\left( 1 \right) = 2\]

 

Q 36.6 | Page 84

Solve the following initial value problem:
(y4 − 2x3 y) dx + (x4 − 2xy3) dy = 0, y (1) = 1

Q 36.7 | Page 84

Solve the following initial value problem:
x (x2 + 3y2) dx + y (y2 + 3x2) dy = 0, y (1) = 1

Q 36.8 | Page 84

Solve the following initial value problem:
\[\left\{ x \sin^2 \left( \frac{y}{x} \right) - y \right\}dx + x dy = 0, y\left( 1 \right) = \frac{\pi}{4}\]

Q 36.9 | Page 84

Solve the following initial value problem:
\[x\frac{dy}{dx} - y + x \sin\left( \frac{y}{x} \right) = 0, y\left( 2 \right) = x\]

Q 37 | Page 84

Find the particular solution of the differential equation x cos\[\left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x\], given that when x = 1, \[y = \frac{\pi}{4}\]

Q 38 | Page 84

Find the particular solution of the differential equation \[\left( x - y \right)\frac{dy}{dx} = x + 2y\], given that when x = 1, y = 0.

Q 39 | Page 84

Find the particular solution of the differential equation \[\frac{dy}{dx} = \frac{xy}{x^2 + y^2}\] given that y = 1 when x = 0.

 

Q 40 | Page 84

Show that the family of curves for which \[\frac{dy}{dx} = \frac{x^2 + y^2}{2xy}\], is given by \[x^2 - y^2 = Cx\]

Chapter 22: Differential Equations solutions [Pages 106 - 108]

Q 1 | Page 106
\[\frac{dy}{dx} + 2y = e^{3x}\]
Q 2 | Page 106
\[4\frac{dy}{dx} + 8y = 5 e^{- 3x}\]
Q 3 | Page 106
\[\frac{dy}{dx} + 2y = 6 e^x\]
Q 4 | Page 106
\[\frac{dy}{dx} + y = e^{- 2x}\]
Q 5 | Page 106
\[x\frac{dy}{dx} = x + y\]
Q 6 | Page 106
\[\frac{dy}{dx} + 2y = 4x\]
Q 7 | Page 106
\[x\frac{dy}{dx} + y = x e^x\]
Q 8 | Page 106
\[\frac{dy}{dx} + \frac{4x}{x^2 + 1}y + \frac{1}{\left( x^2 + 1 \right)^2} = 0\]
Q 9 | Page 106
\[x\frac{dy}{dx} + y = x \log x\]
Q 10 | Page 106
\[x\frac{dy}{dx} - y = \left( x - 1 \right) e^x\]
Q 11 | Page 106
\[\frac{dy}{dx} + \frac{y}{x} = x^3\]
Q 12 | Page 106
\[\frac{dy}{dx} + y = \sin x\]
Q 13 | Page 106
\[\frac{dy}{dx} + y = \cos x\]
Q 14 | Page 106
\[\frac{dy}{dx} + 2y = \sin x\]
Q 15 | Page 106

\[\frac{dy}{dx}\] = y tan x − 2 sin x

Q 16 | Page 106
\[\left( 1 + x^2 \right)\frac{dy}{dx} + y = \tan^{- 1} x\]
Q 17 | Page 106

\[\frac{dy}{dx}\] + y tan x = cos x

Q 18 | Page 106

\[\frac{dy}{dx}\] + y cot x = x2 cot x + 2x

Q 19 | Page 106
\[\frac{dy}{dx} + y \tan x = x^2 \cos^2 x\]
Q 20 | Page 106
\[\left( 1 + x^2 \right)\frac{dy}{dx} + y = e^{tan^{- 1}} x\]
Q 21 | Page 106

x dy = (2y + 2x4 + x2) dx

Q 22 | Page 106
\[\left( 1 + y^2 \right) + \left( x - e^{tan^{- 1}} y \right)\frac{dy}{dx} = 0\]
Q 23 | Page 106
\[y^2 \frac{dx}{dy} + x - \frac{1}{y} = 0\]

 

Q 24 | Page 106
\[\left( 2x - 10 y^3 \right)\frac{dy}{dx} + y = 0\]
Q 25 | Page 106

(x + tan y) dy = sin 2y dx

Q 26 | Page 106

dx + xdy = e−y sec2 y dy

Q 27 | Page 106

\[\frac{dy}{dx}\] = y tan x − 2 sin x

Q 29 | Page 106

\[\frac{dy}{dx}\] + y cos x = sin x cos x

Q 30 | Page 106
\[\left( \sin x \right)\frac{dy}{dx} + y \cos x = 2 \sin^2 x \cos x\]
Q 31 | Page 106
\[\left( x^2 - 1 \right)\frac{dy}{dx} + 2\left( x + 2 \right)y = 2\left( x + 1 \right)\]
Q 32 | Page 106
\[x\frac{dy}{dx} + 2y = x \cos x\]
Q 33 | Page 106
\[\frac{dy}{dx} - y = x e^x\]
Q 34 | Page 106
\[\frac{dy}{dx} + 2y = x e^{4x}\]
Q 35 | Page 106

Solve the differential equation \[\left( x + 2 y^2 \right)\frac{dy}{dx} = y\], given that when x = 2, y = 1.

Q 36.01 | Page 107

Find one-parameter families of solution curves of the following differential equation: (or Solve the following differential equation)
\[\frac{dy}{dx} + 3y = e^{mx}\], m is a given real number

 

Q 36.02 | Page 107

Find one-parameter families of solution curves of the following differential equation: (or Solve the following differential equation)
\[\frac{dy}{dx} - y = \cos 2x\]

Q 36.03 | Page 107

Find one-parameter families of solution curves of the following differential equation: (or Solve the following differential equation)
\[x\frac{dy}{dx} - y = \left( x + 1 \right) e^{- x}\]

Q 36.04 | Page 107

Find one-parameter families of solution curves of the following differential equation: (or Solve the following differential equation)

\[x\frac{dy}{dx} + y = x^4\]
Q 36.05 | Page 107

Find one-parameter families of solution curves of the following differential equation: (or Solve the following differential equation)

\[\left( x \log x \right)\frac{dy}{dx} + y = \log x\]
Q 36.06 | Page 107

Find one-parameter families of solution curves of the following differential equation: (or Solve the following differential equation)

\[\frac{dy}{dx} - \frac{2xy}{1 + x^2} = x^2 + 2\]
Q 36.07 | Page 107

Find one-parameter families of solution curves of the following differential equation: (or Solve the following differential equation)

\[\frac{dy}{dx} + y \cos x = e^{\sin x} \cos x\]
Q 36.08 | Page 107

Find one-parameter families of solution curves of the following differential equation: (or Solve the following differential equation)

\[\left( x + y \right)\frac{dy}{dx} = 1\]
Q 36.09 | Page 107

Find one-parameter families of solution curves of the following differential equation: (or Solve the following differential equation)

\[\frac{dy}{dx} \cos^2 x = \tan x - y\]
Q 36.1 | Page 107

Find one-parameter families of solution curves of the following differential equation: (or Solve the following differential equation)

\[e^{- y} \sec^2 y dy = dx + x dy\]
Q 36.11 | Page 107

Find one-parameter families of solution curves of the following differential equation: (or Solve the following differential equation)

\[x \log x\frac{dy}{dx} + y = 2 \log x\]
Q 36.12 | Page 107

Find one-parameter families of solution curves of the following differential equation: (or Solve the following differential equation)

\[x\frac{dy}{dx} + 2y = x^2 \log x\]
Q 37.01 | Page 107

Solve the following initial value problem:
\[y' + y = e^x , y\left( 0 \right) = \frac{1}{2}\]

Q 37.02 | Page 107

Solve the following initial value problem:
\[x\frac{dy}{dx} - y = \log x, y\left( 1 \right) = 0\]

Q 37.03 | Page 107

Solve the following initial value problem:
\[\frac{dy}{dx} + 2y = e^{- 2x} \sin x, y\left( 0 \right) = 0\]

Q 37.04 | Page 107

Solve the following initial value problem:
\[x\frac{dy}{dx} - y = \left( x + 1 \right) e^{- x} , y\left( 1 \right) = 0\]

Q 37.05 | Page 107

Solve the following initial value problem:
\[\left( 1 + y^2 \right) dx + \left( x - e^{- \tan^{- 1} y} \right) dx = 0, y\left( 0 \right) = 0\]

Q 37.06 | Page 107

Solve the following initial value problem:
\[\frac{dy}{dx} + y \tan x = 2x + x^2 \tan x, y\left( 0 \right) = 1\]

Q 37.07 | Page 107

Solve the following initial value problem:
\[x\frac{dy}{dx} + y = x \cos x + \sin x, y\left( \frac{\pi}{2} \right) = 1\]

Q 37.08 | Page 107

Solve the following initial value problem:
\[\frac{dy}{dx} + y \cot x = 4x\text{ cosec }x, y\left( \frac{\pi}{2} \right) = 0\]

Q 37.09 | Page 107

Solve the following initial value problem:
\[\frac{dy}{dx} + 2y \tan x = \sin x; y = 0\text{ when }x = \frac{\pi}{3}\]

Q 37.1 | Page 107

Solve the following initial value problem:
\[\frac{dy}{dx} - 3y \cot x = \sin 2x; y = 2\text{ when }x = \frac{\pi}{2}\]

Q 37.11 | Page 107

Solve the following initial value problem:
\[\frac{dy}{dx} + y\cot x = 2\cos x, y\left( \frac{\pi}{2} \right) = 0\]

Q 37.12 | Page 107

Solve the following initial value problem:
\[dy = \cos x\left( 2 - y\text{ cosec }x \right)dx\]

Q 37.13 | Page 107

Solve the following initial value problem:
\[\tan x\left( \frac{dy}{dx} \right) = 2x\tan x + x^2 - y; \tan x \neq 0\] \[x = \frac{\pi}{2}\]

Q 38 | Page 107

Find the general solution of the differential equation
\[x\frac{dy}{dx} + 2y = x^2\]

Q 39 | Page 107

Find the general solution of the differential equation
\[\frac{dy}{dx} - y = \cos x\]

Q 40 | Page 107

Solve the differential equation \[\left( y + 3 x^2 \right)\frac{dx}{dy} = x\]

Q 41 | Page 108

Find the particular solution of the differential equation \[\frac{dx}{dy} + x \cot y = 2y + y^2\] cot yy ≠ 0 given that x = 0 when \[y = \frac{\pi}{2}\]

Q 42 | Page 108

Solve the following differential equation: \[\left( \cot^{- 1} y + x \right) dy = \left( 1 + y^2 \right) dx\]

Chapter 22: Differential Equations solutions [Pages 134 - 136]

Q 1 | Page 134

The surface area of a balloon being inflated, changes at a rate proportional to time t. If initially its radius is 1 unit and after 3 seconds it is 2 units, find the radius after time t.

Q 2 | Page 134

A population grows at the rate of 5% per year. How long does it take for the population to double?

Q 3 | Page 134

The rate of growth of a population is proportional to the number present. If the population of a city doubled in the past 25 years, and the present population is 100000, when will the city have a population of 500000?

Q 4 | Page 134

In a culture, the bacteria count is 100000. The number is increased by 10% in 2 hours. In how many hours will the count reach 200000, if the rate of growth of bacteria is proportional to the number present?

Q 5 | Page 134

If the interest is compounded continuously at 6% per annum, how much worth Rs 1000 will be after 10 years? How long will it take to double Rs 1000?

Q 6 | Page 134

The rate of increase in the number of bacteria in a certain bacteria culture is proportional to the number present. Given the number triples in 5 hrs, find how many bacteria will be present after 10 hours. Also find the time necessary for the number of bacteria to be 10 times the number of initial present.

Q 7 | Page 134

The population of a city increases at a rate proportional to the number of inhabitants present at any time t. If the population of the city was 200000 in 1990 and 250000 in 2000, what will be the population in 2010?

Q 8 | Page 134

If the marginal cost of manufacturing a certain item is given by C' (x) = \[\frac{dC}{dx}\] = 2 + 0.15 x. Find the total cost function C (x), given that C (0) = 100.

 
Q 9 | Page 134

A bank pays interest by continuous compounding, that is, by treating the interest rate as the instantaneous rate of change of principal. Suppose in an account interest accrues at 8% per year, compounded continuously. Calculate the percentage increase in such an account over one year.

Q 10 | Page 134

In a simple circuit of resistance R, self inductance L and voltage E, the current i at any time t is given by L \[\frac{di}{dt}\]+ R i = E. If E is constant and initially no current passes through the circuit, prove that \[i = \frac{E}{R}\left\{ 1 - e^{- \left( R/L \right)t} \right\}\]

Q 11 | Page 134

The decay rate of radium at any time t is proportional to its mass at that time. Find the time when the mass will be halved of its initial mass.

Q 12 | Page 134

Experiments show that radium disintegrates at a rate proportional to the amount of radium present at the moment. Its half-life is 1590 years. What percentage will disappear in one year?

Q 13 | Page 135

The slope of the tangent at a point P (x, y) on a curve is \[\frac{- x}{y}\]. If the curve passes through the point (3, −4), find the equation of the curve.

Q 14 | Page 135

Find the equation of the curve which passes through the point (2, 2) and satisfies the differential equation
\[y - x\frac{dy}{dx} = y^2 + \frac{dy}{dx}\]

Q 15 | Page 135

Find the equation of the curve passing through the point \[\left( 1, \frac{\pi}{4} \right)\]  and tangent at any point of which makes an angle tan−1  \[\left( \frac{y}{x} - \cos^2 \frac{y}{x} \right)\] with x-axis.

Q 16 | Page 135

Find the curve for which the intercept cut-off by a tangent on x-axis is equal to four times the ordinate of the point of contact.

 
Q 17 | Page 135

Show that the equation of the curve whose slope at any point is equal to y + 2x and which passes through the origin is y + 2 (x + 1) = 2e2x.

Q 18 | Page 135

The tangent at any point (x, y) of a curve makes an angle tan−1(2x + 3y) with x-axis. Find the equation of the curve if it passes through (1, 2).

Q 19 | Page 135

Find the equation of the curve such that the portion of the x-axis cut off between the origin and the tangent at a point is twice the abscissa and which passes through the point (1, 2).

Q 20 | Page 135

Find the equation to the curve satisfying x (x + 1) \[\frac{dy}{dx} - y\]  = x (x + 1) and passing through (1, 0).

Q 21 | Page 135

Find the equation of the curve which passes through the point (3, −4) and has the slope \[\frac{2y}{x}\]  at any point (x, y) on it.

Q 22 | Page 135

Find the equation of the curve which passes through the origin and has the slope x + 3y− 1 at any point (x, y) on it.

Q 23 | Page 135

At every point on a curve the slope is the sum of the abscissa and the product of the ordinate and the abscissa, and the curve passes through (0, 1). Find the equation of the curve.

Q 24 | Page 135

A curve is such that the length of the perpendicular from the origin on the tangent at any point P of the curve is equal to the abscissa of P. Prove that the differential equation of the curve is \[y^2 - 2xy\frac{dy}{dx} - x^2 = 0\], and hence find the curve.

Q 25 | Page 135

Find the equation of the curve which passes through the point (1, 2) and the distance between the foot of the ordinate of the point of contact and the point of intersection of the tangent with x-axis is twice the abscissa of the point of contact.

Q 26 | Page 135

The normal to a given curve at each point (x, y) on the curve passes through the point (3, 0). If the curve contains the point (3, 4), find its equation.

Q 27 | Page 135

The rate of increase of bacteria in a culture is proportional to the number of bacteria present and it is found that the number doubles in 6 hours. Prove that the bacteria becomes 8 times at the end of 18 hours.

Q 28 | Page 135

Radium decomposes at a rate proportional to the quantity of radium present. It is found that in 25 years, approximately 1.1% of a certain quantity of radium has decomposed. Determine approximately how long it will take for one-half of the original amount of  radium to decompose?

Q 29 | Page 135

Show that all curves for which the slope at any point (x, y) on it is \[\frac{x^2 + y^2}{2xy}\]  are rectangular hyperbola.

Q 30 | Page 136

The slope of the tangent at each point of a curve is equal to the sum of the coordinates of the point. Find the curve that passes through the origin.

Q 31 | Page 136

Find the equation of the curve passing through the point (0, 1) if the slope of the tangent to the curve at each of its point is equal to the sum of the abscissa and the product of the abscissa and the ordinate of the point.

Q 32 | Page 136

The slope of a curve at each of its points is equal to the square of the abscissa of the point. Find the particular curve through the point (−1, 1).

Q 33 | Page 136

Find the equation of the curve that passes through the point (0, a) and is such that at any point (x, y) on it, the product of its slope and the ordinate is equal to the abscissa.

Q 34 | Page 136

The x-intercept of the tangent line to a curve is equal to the ordinate of the point of contact. Find the particular curve through the point (1, 1).

Chapter 22: Differential Equations solutions [Pages 137 - 139]

Q 1 | Page 137

Define a differential equation.

Q 2 | Page 137

Define order of a differential equation.

Q 3 | Page 137

Define degree of a differential equation.

Q 4 | Page 137

Write the differential equation representing the family of straight lines y = Cx + 5, where C is an arbitrary constant.

Q 5 | Page 137

Write the differential equation obtained by eliminating the arbitrary constant C in the equation x2 − y2 = C2.

Q 6 | Page 138

Write the differential equation obtained eliminating the arbitrary constant C in the equation xy = C2.

Q 7 | Page 138

Write the degree of the differential equation
\[a^2 \frac{d^2 y}{d x^2} = \left\{ 1 + \left( \frac{dy}{dx} \right)^2 \right\}^{1/4}\]

Q 8 | Page 138

Write the order of the differential equation
\[1 + \left( \frac{dy}{dx} \right)^2 = 7 \left( \frac{d^2 y}{d x^2} \right)^3\]

Q 9 | Page 138

Write the order and degree of the differential equation
\[y = x\frac{dy}{dx} + a\sqrt{1 + \left( \frac{dy}{dx} \right)^2}\]

Q 10 | Page 138

Write the degree of the differential equation
\[\frac{d^2 y}{d x^2} + x \left( \frac{dy}{dx} \right)^2 = 2 x^2 \log \left( \frac{d^2 y}{d x^2} \right)\]

Q 11 | Page 138

Write the order of the differential equation of the family of circles touching X-axis at the origin.

Q 12 | Page 138

Write the order of the differential equation of all non-horizontal lines in a plane.

Q 13 | Page 138

If sin x is an integrating factor of the differential equation \[\frac{dy}{dx} + Py = Q\], then write the value of P.

Q 15 | Page 138

Write the order of the differential equation whose solution is y = a cos x + b sin x + c e−x.

Q 16 | Page 138

Write the order of the differential equation associated with the primitive y = C1 + C2 ex + C3 e−2x + C4, where C1, C2, C3, C4 are arbitrary constants.

Q 17 | Page 138

What is the degree of the following differential equation?

\[5x \left( \frac{dy}{dx} \right)^2 - \frac{d^2 y}{d x^2} - 6y = \log x\]
Q 18 | Page 138

Write the degree of the differential equation \[\left( \frac{dy}{dx} \right)^4 + 3x\frac{d^2 y}{d x^2} = 0\]

Q 19 | Page 138

Write the degree of the differential equation x \[\left( \frac{d^2 y}{d x^2} \right)^3 + y \left( \frac{dy}{dx} \right)^4 + x^3 = 0\]

 

Q 20 | Page 138

Write the differential equation representing family of curves y = mx, where m is arbitrary constant.

Q 21 | Page 138

Write the degree of the differential equation \[x^3 \left( \frac{d^2 y}{d x^2} \right)^2 + x \left( \frac{dy}{dx} \right)^4 = 0\]

Q 22 | Page 138

Write the degree of the differential equation \[\left( 1 + \frac{dy}{dx} \right)^3 = \left( \frac{d^2 y}{d x^2} \right)^2\]

Q 23 | Page 138

Write the degree of the differential equation \[\frac{d^2 y}{d x^2} + 3 \left( \frac{dy}{dx} \right)^2 = x^2 \log\left( \frac{d^2 y}{d x^2} \right)\]

Q 24 | Page 139

Write the degree of the differential equation \[\left( \frac{d^2 y}{d x^2} \right)^2 + \left( \frac{dy}{dx} \right)^2 = x\sin\left( \frac{dy}{dx} \right)\]

Q 25 | Page 139

Write the order and degree of the differential equation
\[\frac{d^2 y}{d x^2} + \left( \frac{dy}{dx} \right)^\frac{1}{4} + x^\frac{1}{5} = 0\]

Q 26 | Page 139

The degree ofthe differential equation \[\frac{d^2 y}{d x^2} + e^\frac{dy}{dx} = 0\]

Q 27 | Page 139

How many arbitrary constants are there in the general solution of the differential equation of order 3.

Q 28 | Page 139

Write the order of the differential equation representing the family of curves y = ax + a3.

Q 29 | Page 139

Find the sum of the order and degree of the differential equation
\[y = x \left( \frac{dy}{dx} \right)^3 + \frac{d^2 y}{d x^2}\]

Q 30 | Page 139

Find the solution of the differential equation
\[x\sqrt{1 + y^2}dx + y\sqrt{1 + x^2}dy = 0\]

Chapter 22: Differential Equations solutions [Pages 139 - 144]

Q 1 | Page 139

The integrating factor of the differential equation (x log x)
\[\frac{dy}{dx} + y = 2 \log x\], is given by
(a) log (log x)
(b) ex
(c) log x
(d) x

Q 2 | Page 139

The general solution of the differential equation \[\frac{dy}{dx} = \frac{y}{x}\] is
(a) log y = kx
(b) y = kx
(c) xy = k
(d) y = k log x

Q 3 | Page 139

Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y\] sin x = 1, is
(a) sin x
(b) sec x
(c) tan x
(d) cos x

Q 4 | Page 139

The degree of the differential equation
\[\left( \frac{d^2 y}{d x^2} \right)^2 - \left( \frac{dy}{dx} \right) = y^3\]

(a) 1/2
(b) 2
(c) 3
(d) 4
Q 5 | Page 140

The degree of the differential equation

\[\left\{ 5 + \left( \frac{dy}{dx} \right)^2 \right\}^{5/3} = x^5 \left( \frac{d^2 y}{d x^2} \right)\]
(a) 4
(b) 2
(c) 5
(d) 10
Q 6 | Page 140

The general solution of the differential equation \[\frac{dy}{dx} + y \]cot x = cosec x, is
(a) x + y sin x = C
(b) x + y cos x = C
(c) y + x (sin x + cos x) = C
(d) y sin x = x + C 

Q 7 | Page 140

The differential equation obtained on eliminating A and B from y = A cos ωt + B sin ωt, is
(a) y" + y' = 0
(b) y" − ω2 y = 0
(c) y" = −ω2 y
(d) y" + y = 0

Q 8 | Page 140

The equation of the curve whose slope is given by \[\frac{dy}{dx} = \frac{2y}{x}; x > 0, y > 0\] and which passes through the point (1, 1) is
(a) x2 = y
(b) y2 = x
(c) x2 = 2y
(d) y2 = 2x

Q 9 | Page 140

The order of the differential equation whose general solution is given by
y = c1 cos (2x + c2) − (c3 + c4) ax + c5 + c6 sin (x − c7) is
(a) 3
(b) 4
(c) 5
(d) 2

Q 10 | Page 140

The solution of the differential equation \[\frac{dy}{dx} = \frac{ax + g}{by + f}\] represents a circle when
(a) a = b
(b) a = −b
(c) a = −2b
(d) a = 2b

Q 11 | Page 140

The solution of the differential equation
\[\frac{dy}{dx} + \frac{2y}{x} = 0\] with y(1) = 1 is given by
(a) \[y = \frac{1}{x^2}\]
(b) \[x = \frac{1}{y^2}\]
(c) \[x = \frac{1}{y}\]
(d) \[y = \frac{1}{x}\]

Q 12 | Page 140

The solution of the differential equation \[\frac{dy}{dx} - \frac{y\left( x + 1 \right)}{x} = 0\] is given by
(a) y = xex + C
(b) x = yex
(c) y = x + C
(d) xy = ex + C

Q 13 | Page 140

The order of the differential equation satisfying
\[\sqrt{1 - x^4} + \sqrt{1 - y^4} = a\left( x^2 - y^2 \right)\]
(a) 1
(b) 2
(c) 3
(d) 4

Q 14 | Page 140

The solution of the differential equation y1 y3 = y22 is
(a) x = C1 eC2y + C3
(b) y = C1 eC2x + C3
(c) 2x = C1 eC2y + C3
(d) none of these

Q 15 | Page 140

The general solution of the differential equation \[\frac{dy}{dx} + y\] g' (x) = g (x) g' (x), where g (x) is a given function of x, is
(a) g (x) + log {1 + y + g (x)} = C
(b) g (x) + log {1 + y − g (x)} = C
(c) g (x) − log {1 + y − g (x)} = C
(d) none of these

Q 16 | Page 140

The solution of the differential equation \[\frac{dy}{dx} = 1 + x + y^2 + x y^2 , y\left( 0 \right) = 0\] is
(a) \[y^2 = \exp\left( x + \frac{x^2}{2} \right) - 1\]
(b) \[y^2 = 1 + C \exp\left( x + \frac{x^2}{2} \right)\]
(c) y = tan (C + x + x2)
(d) \[y = \tan\left( x + \frac{x^2}{2} \right)\] 

Q 17 | Page 141

The differential equation of the ellipse \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = C\]
(a) \[\frac{y "}{y'} + \frac{y'}{y} - \frac{1}{x} = 0\]
(b) \[\frac{y "}{y'} + \frac{y'}{y} + \frac{1}{x} = 0\]
(c) \[\frac{y "}{y'} - \frac{y'}{y} - \frac{1}{x} = 0\]
(d) none of these

Q 18 | Page 141

Solution of the differential equation \[\frac{dy}{dx} + \frac{y}{x}\]
(a) x (y + cos x) = sin x + C
(b) x (y − cos x) = sin x + C
(c) x (y + cos x) = cos x + C
(d) none of these

Q 19 | Page 141

The equation of the curve satisfying the differential equation y (x + y3) dx = x (y3 − x) dyand passing through the point (1, 1) is
(a) y3 − 2x + 3x2 y = 0
(b) y3 + 2x + 3x2 y = 0
(c) y3 + 2x −3x2 y = 0
(d) none of these

Q 20 | Page 141

The solution of the differential equation \[2x\frac{dy}{dx} - y = 3\] represents
(a) circles
(b) straight lines
(c) ellipses
(d) parabolas

Q 21 | Page 141

The solution of the differential equation \[x\frac{dy}{dx} = y + x \tan\frac{y}{x}\]
(a) \[\sin\frac{x}{y} = x + C\]
c(c) \[\sin\frac{x}{y} = Cy\]
(d) \[\sin\frac{y}{x} = Cy\]

Q 22 | Page 141

The differential equation satisfied by ax2 + by2 = 1 is
(a) xyy2 + y12 + yy1 = 0
(b) xyy2 + xy12 − yy1 = 0
(c) xyy2 − xy12 + yy1 = 0
(d) none of these

Q 23 | Page 141

The differential equation which represents the family of curves y = eCx is
(a) y1 = C2 y
(b) xy1 − ln y = 0
(c) x ln y = yy1
(d) y ln y = xy1

Q 24 | Page 141

Which of the following transformations reduce the differential equation \[\frac{dz}{dx} + \frac{z}{x}\log z = \frac{z}{x^2} \left( \log z \right)^2\] into the form \[\frac{du}{dx} + P\left( x \right) u = Q\left( x \right)\]
(a) u = log x
(b) u = ez
(c) u = (log z)−1
(d) u = (log z)2

Q 25 | Page 141

If m and n are the order and degree of the differential equation \[\left( y_2 \right)^5 + \frac{4 \left( y_2 \right)^3}{y_3} + y_3 = x^2 - 1\]
(a) m = 3, n = 3
(b) m = 3, n = 2
(c) m = 3, n = 5
(d) m = 3, n = 1

Q 26 | Page 141

If m and n are the order and degree of the differential equation \[\left( y_2 \right)^5 + \frac{4 \left( y_2 \right)^3}{y_3} + y_3 = x^2 - 1\]
(a) m = 3, n = 3
(b) m = 3, n = 2
(c) m = 3, n = 5
(d) m = 3, n = 1

Q 27 | Page 142

The solution of the differential equation \[\frac{dy}{dx} + 1 = e^{x + y}\], is
(a) (x + y) ex + y = 0
(b) (x + C) ex + y = 0
(c) (x − C) ex + y = 1
(d) (x − C) ex + y + 1 =0

Q 28 | Page 142

The solution of x2 + y \[\frac{dy}{dx}\]= 4, is
(a) x2 + y2 = 12x + C
(b) x2 + y2 = 3x + C
(c) x3 + y3 = 3x + C
(d) x3 + y3 = 12x + C

Q 29 | Page 142

The family of curves in which the sub tangent at any point of a curve is double the abscissae, is given by
(a) x = Cy2
(b) y = Cx2
(c) x2 = Cy2
(d) y = Cx

Q 30 | Page 142

The solution of the differential equation x dx + y dy = x2 y dy − y2 x dx, is
(a) x2 − 1 = C (1 + y2)
(b) x2 + 1 = C (1 − y2)
(c) x3 − 1 = C (1 + y3)
(d) x3 + 1 = C (1 − y3)

Q 31 | Page 142

The solution of the differential equation (x2 + 1) \[\frac{dy}{dx}\] + (y2 + 1) = 0, is
(a) y = 2 + x2
(b) \[y = \frac{1 + x}{1 - x}\]
(c) y = x (x − 1)
(d) \[y = \frac{1 - x}{1 + x}\]

Q 32 | Page 142

The differential equation \[x\frac{dy}{dx} - y = x^2\], has the general solution 
(a) y − x3 = 2cx
(b) 2y − x3 = cx
(c) 2y + x2 = 2cx
(d) y + x2 = 2cx

Q 33 | Page 142

The solution of the differential equation \[\frac{dy}{dx} - ky = 0, y\left( 0 \right) = 1\] approaches to zero when x → ∞, if
(a) k = 0
(b) k > 0
(c) k < 0
(d) none of these

Q 34 | Page 142

The solution of the differential equation \[\left( 1 + x^2 \right)\frac{dy}{dx} + 1 + y^2 = 0\], is
(a) tan1 x − tan−1 y = tan−1 C
(b) tan−1 y − tan−1 x = tan−1 C
(c) tan−1 y ± tan−1 x = tan C
(d) tan−1 y + tan−1 x = tan−1 C

 

Q 35 | Page 142

The solution of the differential equation \[\frac{dy}{dx} = \frac{x^2 + xy + y^2}{x^2}\], is
(a) \[\tan^{- 1} \left( \frac{x}{y} \right) = \log y + C\]
(b) \[\tan^{- 1} \left( \frac{y}{x} \right) = \log x + C\]
(c) \[\tan^{- 1} \left( \frac{x}{y} \right) = \log x + C\]
(d) \[\tan^{- 1} \left( \frac{y}{x} \right) = \log y + C\]

Q 36 | Page 142

The differential equation
\[\frac{dy}{dx} + Py = Q y^n , n > 2\] can be reduced to linear form by substituting
(a) z = yn −1
(b) z = yn
(c) z = yn + 1
(d) z = y1 − n

Q 37 | Page 142

If p and q are the order and degree of the differential equation \[y\frac{dy}{dx} + x^3 \frac{d^2 y}{d x^2} + xy\] = cos x, then
(a) p < q
(b) p = q
(c) p > q
(d) none of these

Q 38 | Page 143

Which of the following is the integrating factor of (x log x) \[\frac{dy}{dx} + y\]
= 2 log x?
(a) x
(b) ex
(c) log x
(d) log (log x)

 

Q 39 | Page 143

What is integrating factor of \[\frac{dy}{dx}\] + y sec x = tan x?
(a) sec x + tan x
(b) log (sec x + tan x)
(c) esec x
(d) sec x

Q 40 | Page 143

Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y \sin x = 1\], is
(a) cos x
(b) tan x
(c) sec x
(d) sin x

Q 41 | Page 143

The degree of the differential equation \[\left( \frac{d^2 y}{d x^2} \right)^3 + \left( \frac{dy}{dx} \right)^2 + \sin\left( \frac{dy}{dx} \right) + 1 = 0\], is
(a) 3
(b) 2
(c) 1
(d) not defined

Q 42 | Page 143

The order of the differential equation \[2 x^2 \frac{d^2 y}{d x^2} - 3\frac{dy}{dx} + y = 0\], is
(a) 2
(b) 1
(c) 0
(d) not defined

Q 43 | Page 143

The number of arbitrary constants in the general solution of differential equation of fourth order is
(a) 0
(b) 2
(c) 3
(d) 4

Q 44 | Page 143

The number of arbitrary constants in the particular solution of a differential equation of third order is
(a) 3
(b) 2
(c) 1
(d) 0

Q 45 | Page 143

Which of the following differential equations has y = C1 ex + C2 ex as the general solution?
(a) \[\frac{d^2 y}{d x^2} + y = 0\]
(b) \[\frac{d^2 y}{d x^2} - y = 0\]
(c) \[\frac{d^2 y}{d x^2} + 1 = 0\]
(d) \[\frac{d^2 y}{d x^2} - 1 = 0\]

Q 46 | Page 143

Which of the following differential equations has y = x as one of its particular solution?
(a) \[\frac{d^2 y}{d x^2} - x^2 \frac{dy}{dx} + xy = x\]
(b) \[\frac{d^2 y}{d x^2} + x\frac{dy}{dx} + xy = x\]
(c) \[\frac{d^2 y}{d x^2} - x^2 \frac{dy}{dx} + xy = 0\]
(d) \[\frac{d^2 y}{d x^2} + x\frac{dy}{dx} + xy = 0\]

Q 47 | Page 143

The general solution of the differential equation \[\frac{dy}{dx} = e^{x + y}\], is
(a) ex + e−y = C
(b) ex + ey = C
(c) ex + ey = C
(d) e−x + e−y = C

Q 48 | Page 143

A homogeneous differential equation of the form \[\frac{dx}{dy} = h\left( \frac{x}{y} \right)\] can be solved by making the substitution
(a) y = vx
(b) v = yx
(c) x = vy
(d) x = v

Q 49 | Page 143

Which of the following is a homogeneous differential equation?
(a) (4x + 6y + 5) dy − (3y + 2x + 4) dx = 0
(b) xy dx − (x3 + y3) dy = 0
(c) (x3 + 2y2) dx + 2xy dy = 0
(d) y2 dx + (x2 − xy − y2) dy = 0

Q 50 | Page 143

The integrating factor of the differential equation \[x\frac{dy}{dx} - y = 2 x^2\]
(a) e−x
(b) ey
(c) \[\frac{1}{x}\]
(d) x

 

Q 51 | Page 144

The integrating factor of the differential equation \[\left( 1 - y^2 \right)\frac{dx}{dy} + yx = ay\left( - 1 < y < 1 \right)\]
(a) \[\frac{1}{y^2 - 1}\]
(b) \[\frac{1}{\sqrt{y^2 - 1}}\]
(c) \[\frac{1}{1 - y^2}\]
(d) \[\frac{1}{\sqrt{1 - y^2}}\]

Q 52 | Page 144

The general solution of the differential equation \[\frac{y dx - x dy}{y} = 0\], is
(a) xy = C
(b) x = Cy2
(c) y = Cx
(d) y = Cx2

 

Q 53 | Page 144

The general solution of a differential equation of the type \[\frac{dx}{dy} + P_1 x = Q_1\] is
(a) \[y e^\int P_1 dy = \int\left\{ Q_1 e^\int P_1 dy \right\}dy + C\]
(b) \[y e^\int P_1 dy = \int\left\{ Q_1 e^\int P_1 dy \right\}dy + C\]
(c) \[x e^\int P_1 dy = \int\left\{ Q_1 e^\int P_1 dy \right\}dy + C\]
(d) \[x e^\int P_1 dy = \int\left\{ Q_1 e^\int P_1 dy \right\}dy + C\]

Q 54 | Page 144

The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is
(a) x ey + x2 = C
(b) x ey + y2 = C
(c) y ex + x2 = C
(d) y ey + x2 = C

Chapter 22: Differential Equations Extra questions

The rate of increase in the number of bacteria in a certain bacteria culture is proportional to the number present. Given the number triples in 5 hrs, find how many bacteria will be present after 10 hours. Also find the time necessary for the number of bacteria to be 10 times the number of initial present.

Chapter 22: Differential Equations

RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

RD Sharma solutions for Class 12 Mathematics chapter 22 - Differential Equations

RD Sharma solutions for Class 12 Maths chapter 22 (Differential Equations) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12 Mathematics chapter 22 Differential Equations are Basic Concepts of Differential Equation, Order and Degree of a Differential Equation, General and Particular Solutions of a Differential Equation, Formation of a Differential Equation Whose General Solution is Given, Differential Equations with Variables Separable, Homogeneous Differential Equations, Solutions of Linear Differential Equation, Linear Differential Equations, Procedure to Form a Differential Equation that Will Represent a Given Family of Curves.

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