R.D. Sharma solutions for Class 12 Mathematics chapter 22 - Differential Equations

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Show that y = Ae^Bx is a solution of the differential equation

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 = e^x + 1

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

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

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

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 = e^x + e^2x

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 = xe^x + e^x

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

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

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

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

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

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

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

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

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.

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.

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

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

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

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

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.

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).

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

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

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

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

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

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

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

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

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

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

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

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

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, \[y = \frac{\pi}{4}\]

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

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

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

\[\frac{dy}{dx}\] = y tan x − 2 sin x

\[\frac{dy}{dx}\] + y tan x = cos x

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

\[\frac{dy}{dx}\] = y tan x − 2 sin x

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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 \[i = \frac{E}{R}\left\{ 1 - e^{- \left( R/L \right)t} \right\}\]

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.

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

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.

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

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.

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.

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.

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

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

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

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

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

What is the degree of the following differential equation?

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

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

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

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

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

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

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

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

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

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

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) e^x
(c) log x
(d) x

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

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

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

The degree of the differential equation

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 

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

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

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

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

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) \[y = \frac{1}{x}\]

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

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

The solution of the differential equation y₁ y₃ = y₂² is
(a) x = C₁ e^C₂y + C₃
(b) y = C₁ e^C₂x + C₃
(c) 2x = C₁ e^C₂y + C₃
(d) none of these

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

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) \[y = \tan\left( x + \frac{x^2}{2} \right)\] 

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

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

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

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

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) \[\sin\frac{y}{x} = Cy\]

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

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

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 = e^z
(c) u = (log z)⁻¹
(d) u = (log z)²

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

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

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

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

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

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)

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) \[y = \frac{1 - x}{1 + x}\]

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

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

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

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) \[\tan^{- 1} \left( \frac{y}{x} \right) = \log y + C\]

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

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

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

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) e^sec x
(d) sec x

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

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

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

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

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

Which of the following differential equations has y = C₁ e^x + C₂ e^−x 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) \[\frac{d^2 y}{d x^2} - 1 = 0\]

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) \[\frac{d^2 y}{d x^2} + x\frac{dy}{dx} + xy = 0\]

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

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

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

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

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) \[\frac{1}{\sqrt{1 - y^2}}\]

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

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) \[x e^\int P_1 dy = \int\left\{ Q_1 e^\int P_1 dy \right\}dy + C\]

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