Chapters
Chapter 2 - Functions
Chapter 3 - Binary Operations
Chapter 4 - Inverse Trigonometric Functions
Chapter 5 - Algebra of Matrices
Chapter 6 - Determinants
Chapter 7 - Adjoint and Inverse of a Matrix
Chapter 8 - Solution of Simultaneous Linear Equations
Chapter 9 - Continuity
Chapter 10 - Differentiability
Chapter 11 - Differentiation
Chapter 12 - Higher Order Derivatives
Chapter 13 - Derivative as a Rate Measurer
Chapter 14 - Differentials, Errors and Approximations
Chapter 15 - Mean Value Theorems
Chapter 16 - Tangents and Normals
Chapter 17 - Increasing and Decreasing Functions
Chapter 18 - Maxima and Minima
Chapter 19 - Indefinite Integrals
Chapter 20 - Definite Integrals
Chapter 21 - Areas of Bounded Regions
Chapter 22 - Differential Equations
Chapter 23 - Algebra of Vectors
Chapter 24 - Scalar Or Dot Product
Chapter 25 - Vector or Cross Product
Chapter 26 - Scalar Triple Product
Chapter 27 - Direction Cosines and Direction Ratios
Chapter 28 - Straight Line in Space
Chapter 29 - The Plane
Chapter 30 - Linear programming
Chapter 31 - Probability
Chapter 32 - Mean and Variance of a Random Variable
Chapter 33 - Binomial Distribution
Chapter 22 - Differential Equations
Pages 4 - 5
(xy2 + x) dx + (y − x2y) dy = 0
(y'')2 + (y')3 + sin y = 0
Pages 16 - 17
Form the differential equation of the family of curves represented by y2 = (x − c)3.
Form the differential equation corresponding to y = emx by eliminating m.
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
Find the differential equation of the family of curves y = Ae2x + Be−2x, where A and B are arbitrary constants.
Find the differential equation of the family of curves, x = A cos nt + B sin nt, where A and B are arbitrary constants.
Form the differential equation corresponding to y2 = a (b − x2) by eliminating a and b.
Form the differential equation corresponding to y2 − 2ay + x2 = a2 by eliminating a.
Form the differential equation corresponding to (x − a)2 + (y − b)2 = r2 by eliminating a and b.
Find the differential equation of all the circles which pass through the origin and whose centres lie on y-axis.
Find the differential equation of all the circles which pass through the origin and whose centres lie on x-axis.
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.
Find the differential equation of all the parabolas with latus rectum '4a' and whose axes are parallel to x-axis.
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 = eax
Form the differential equation representing the family of ellipses having centre at the origin and foci on x-axis.
Form the differential equation of the family of hyperbolas having foci on x-axis and centre at the origin.
Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.
Pages 24 - 25
Show that y = bex + ce2x is a solution of the differential equation,
\[\frac{d^2 y}{d x^2} - 3\frac{dy}{dx} + 2y = 0\]
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 = AeBx 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 \[\frac{d^3 y}{d x^3} = 6a\].
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) \[\frac{dy}{dx}\].
Show that y = ex (A cos x + B sin x) is the solution of the differential equation \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + 2y = 0\]
Verify that y = cx + 2c2 is a solution of the differential equation
Verify that y = − x − 1 is a solution of the differential equation (y − x) dy − (y2 − x2) dx = 0.
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:
Differential equation | Function |
\[x\frac{dy}{dx} = y\]
|
y = ax |
For the following differential equation verify that the accompanying function is a solution:
Differential equation | Function |
\[x + y\frac{dy}{dx} = 0\]
|
\[y = \pm \sqrt{a^2 - x^2}\]
|
For the following differential equation verify that the accompanying function is a solution:
Differential equation | Function |
\[x\frac{dy}{dx} + y = y^2\]
|
\[y = \frac{a}{x + a}\]
|
For the following differential equation verify that the accompanying function is a solution:
Differential equation | Function |
\[x^3 \frac{d^2 y}{d x^2} = 1\]
|
\[y = ax + b + \frac{1}{2x}\]
|
For the following differential equation verify that the accompanying function is a solution:
Differential equation | Function |
\[y = \left( \frac{dy}{dx} \right)^2\]
|
\[y = \frac{1}{4} \left( x \pm a \right)^2\]
|
Page 28
Differential equation \[x\frac{dy}{dx} = 1, y\left( 1 \right) = 0\]
Function y = log x
Differential equation \[\frac{dy}{dx} = y, y\left( 0 \right) = 1\]
Function y = ex
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 = ex + 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 = ex + 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 = ex + e2x
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
Page 34
(sin x + cos x) dy + (cos x − sin x) dx = 0
C' (x) = 2 + 0.15 x ; C(0) = 100
Page 38
Pages 55 - 57
(1 + x2) dy = xy dx
xy (y + 1) dy = (x2 + 1) dx
x cos y dy = (xex log x + ex) dx
(ey + 1) cos x dx + ey sin x dy = 0
x cos2 y dx = y cos2 x dy
xy dy = (y − 1) (x + 1) dx
(1 − x2) dy + xy dx = xy2 dx
tan y dx + sec2 y tan x dy = 0
(1 + x) (1 + y2) dx + (1 + y) (1 + x2) dy = 0
tan y \[\frac{dy}{dx}\] = sin (x + y) + sin (x − y)
y (1 + ex) dy = (y + 1) ex dx
(y + xy) dx + (x − xy2) dy = 0
(y2 + 1) dx − (x2 + 1) dy = 0
dy + (x + 1) (y + 1) dx = 0
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 particular solution of edy/dx = x + 1, given that y = 3, when x = 0.
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).
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.
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).
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).
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.
Page 66
(x + y) (dx − dy) = dx + dy
Pages 83 - 84
x2 dy + y (x + y) dx = 0
(x2 − y2) dx − 2xy dy = 0
y ex/y dx = (xex/y + y) dy
\[x^2 \frac{dy}{dx} = x^2 + xy + y^2 \]
(y2 − 2xy) dx = (x2 − 2xy) dy
2xy dx + (x2 + 2y2) dy = 0
3x2 dy = (3xy + y2) dx
(x + 2y) dx − (2x − y) dy = 0
Solve the following differential equations:
\[\frac{dy}{dx} = \frac{y}{x}\left\{ \log y - \log x + 1 \right\}\]
y2 dx + (x2 − xy + y2) dy = 0
y2 dx + (x2 − xy + y2) dy = 0
(x2 − 2xy) dy + (x2 − 3xy + 2y2) dx = 0
(x2 + 3xy + y2) dx − x2 dy = 0
(2x2 y + y3) dx + (xy2 − 3x3) dy = 0
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 \[x^2 - y^2 = Cx\]
Pages 106 - 108
\[\frac{dy}{dx}\] = y tan x − 2 sin x
\[\frac{dy}{dx}\] + y tan x = cos x
\[\frac{dy}{dx}\] + y cot x = x2 cot x + 2x
x dy = (2y + 2x4 + x2) dx
(x + tan y) dy = sin 2y dx
dx + xdy = e−y sec2 y dy
\[\frac{dy}{dx}\] = 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: \[\left( \cot^{- 1} y + x \right) dy = \left( 1 + y^2 \right) dx\]
Pages 134 - 136
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.
A population grows at the rate of 5% per year. How long does it take for the population to double?
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?
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 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?
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.
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?
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.
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.
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 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.
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?
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 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.
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.
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).
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).
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.
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.
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.
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.
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.
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.
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.
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?
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.
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.
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.
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).
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.
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).
Pages 137 - 139
Define a differential equation.
Define order of a differential equation.
Define degree of a differential equation.
Write the differential equation representing the family of straight lines y = Cx + 5, where C is an arbitrary constant.
Write the differential equation obtained by eliminating the arbitrary constant C in the equation x2 − y2 = C2.
Write the differential equation obtained eliminating the arbitrary constant C in the equation xy = C2.
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
\[\frac{d^2 y}{d x^2} + x \left( \frac{dy}{dx} \right)^2 = 2 x^2 \log \left( \frac{d^2 y}{d x^2} \right)\]
Write the order of the differential equation of the family of circles touching X-axis at the origin.
Write the order of the differential equation of all non-horizontal lines in a plane.
If sin x is an integrating factor of the differential equation \[\frac{dy}{dx} + Py = Q\], then write the value of P.
Write the order of the differential equation whose solution is y = a cos x + b sin x + c e−x.
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.
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 differential equation representing family of curves y = mx, where m is arbitrary constant.
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 \[\frac{d^2 y}{d x^2} + e^\frac{dy}{dx} = 0\]
How many arbitrary constants are there in the general solution of the differential equation of order 3.
Write the order of the differential equation representing the family of curves y = ax + a3.
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
\[x\sqrt{1 + y^2}dx + y\sqrt{1 + x^2}dy = 0\]
Pages 139 - 144
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
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
\[\left( \frac{d^2 y}{d x^2} \right)^2 - \left( \frac{dy}{dx} \right) = y^3\]
(b) 2
(c) 3
(d) 4
The degree of the differential equation
(a) 4
(b) 2
(c) 5
(d) 10
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) ax + 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 = xex + C
(b) x = yex
(c) y = x + C
(d) xy = ex + 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 eC2y + C3
(b) y = C1 eC2x + C3
(c) 2x = C1 eC2y + 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 = eCx 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 = ez
(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) ex + y = 0
(b) (x + C) ex + y = 0
(c) (x − C) ex + y = 1
(d) (x − C) ex + 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 = yn −1
(b) z = yn
(c) z = yn + 1
(d) z = y1 − 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) ex
(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) esec 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 ex + 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) ex + e−y = C
(b) ex + ey = C
(c) e−x + ey = 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 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
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.
Textbook solutions for Class 12
R.D. Sharma solutions for Class 12 Mathematics chapter 22 - Differential Equations
R.D. Sharma solutions for Class 12 Mathematics chapter 22 (Differential Equations) include all questions with solution and detail explanation from Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session). 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 created 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 Procedure to Form a Differential Equation that Will Represent a Given Family of Curves, Linear Differential Equations, Solutions of Linear Differential Equation, Homogeneous Differential Equations, Differential Equations with Variables Separable, Formation of a Differential Equation Whose General Solution is Given, General and Particular Solutions of a Differential Equation, Order and Degree of a Differential Equation, Basic Concepts of Differential Equation.
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