RD 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 y = ae^2x + be^−x is a solution of the differential equation \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} - 2y = 0\]

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 \[x\frac{dy}{dx} + \cot y = 0\] given that \[y = \frac{\pi}{4}\], when \[x=\sqrt{2}\]

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:
\[\frac{dy}{dx} = \frac{y}{x}\left\{ \log y - \log x + 1 \right\}\]

y² dx + (x² − xy + y²) 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 \[x^2 - y^2 = Cx\]

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

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:-

\[\frac{dy}{dx} + 3y = e^{mx}\], m is a given real number.

Find one-parameter families of solution curves of the following differential equation:-

\[\frac{dy}{dx} - y = \cos 2x\]

Find one-parameter families of solution curves of 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:-

\[x\frac{dy}{dx} + y = x^4\]

Find one-parameter families of solution curves of the following differential equation:-

\[\left( x \log x \right)\frac{dy}{dx} + y = \log x\]

Find one-parameter families of solution curves of the following differential equation:-

\[\frac{dy}{dx} - \frac{2xy}{1 + x^2} = x^2 + 2\]

Find one-parameter families of solution curves of the following differential equation:-

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

Find one-parameter families of solution curves of the following differential equation:-

\[\left( x + y \right)\frac{dy}{dx} = 1\]

Find one-parameter families of solution curves of the following differential equation:-

\[\frac{dy}{dx} \cos^2 x = \tan x - y\]

Find one-parameter families of solution curves of the following differential equation:-

\[e^{- y} \sec^2 y dy = dx + x dy\]

Find one-parameter families of solution curves of the following differential equation:-

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

Find one-parameter families of solution curves of the following differential equation:-

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

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:-
\[\tan x\left( \frac{dy}{dx} \right) = 2x\tan x + x^2 - y; \tan x \neq 0\] given that y = 0 when \[x = \frac{\pi}{2}\]

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 of the 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

The general solution of the differential equation \[\frac{dy}{dx} = \frac{y}{x}\] is

Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y\] sin x = 1, is

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

The degree of the differential equation \[\left\{ 5 + \left( \frac{dy}{dx} \right)^2 \right\}^{5/3} = x^5 \left( \frac{d^2 y}{d x^2} \right)\], is

The general solution of the differential equation \[\frac{dy}{dx} + y \] cot x = cosec x, is

The differential equation obtained on eliminating A and B from y = A cos ωt + B sin ωt, is

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

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

The solution of the differential equation \[\frac{dy}{dx} = \frac{ax + g}{by + f}\] represents a circle when

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

The solution of the differential equation \[\frac{dy}{dx} - \frac{y\left( x + 1 \right)}{x} = 0\] is given by

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

The solution of the differential equation y₁ y₃ = y₂² is

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

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

The differential equation of the ellipse \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = C\] is

Solution of the differential equation \[\frac{dy}{dx} + \frac{y}{x}=\sin x\] is

The equation of the curve satisfying the differential equation y (x + y³) dx = x (y³ − x) dy and passing through the point (1, 1) is

The solution of the differential equation \[2x\frac{dy}{dx} - y = 3\] represents

The solution of the differential equation \[x\frac{dy}{dx} = y + x \tan\frac{y}{x}\], is

The differential equation satisfied by ax² + by² = 1 is

The differential equation which represents the family of curves y = e^Cx is

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 \[\frac{du}{dx} + P\left( x \right) u = Q\left( x \right)\]

The solution of the differential equation \[\frac{dy}{dx} = \frac{y}{x} + \frac{\phi\left( \frac{y}{x} \right)}{\phi'\left( \frac{y}{x} \right)}\] is

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

The solution of the differential equation \[\frac{dy}{dx} + 1 = e^{x + y}\], is

The solution of x² + y²  \[\frac{dy}{dx}\]= 4, is

The family of curves in which the sub tangent at any point of a curve is double the abscissae, is given by

The solution of the differential equation x dx + y dy = x² y dy − y² x dx, is

The solution of the differential equation (x² + 1) \[\frac{dy}{dx}\] + (y² + 1) = 0, is

The differential equation \[x\frac{dy}{dx} - y = x^2\], has the general solution

The solution of the differential equation \[\frac{dy}{dx} - ky = 0, y\left( 0 \right) = 1\] approaches to zero when x → ∞, if

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

The solution of the differential equation \[\frac{dy}{dx} = \frac{x^2 + xy + y^2}{x^2}\], is

The differential equation
\[\frac{dy}{dx} + Py = Q y^n , n > 2\] can be reduced to linear form by substituting

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

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

What is integrating factor of \[\frac{dy}{dx}\] + y sec x = tan x?

Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y \sin x = 1\], is

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

The order of the differential equation \[2 x^2 \frac{d^2 y}{d x^2} - 3\frac{dy}{dx} + y = 0\], is

The number of arbitrary constants in the general solution of differential equation of fourth order is

The number of arbitrary constants in the particular solution of a differential equation of third order is

Which of the following differential equations has y = C₁ e^x + C₂ e^−x as the general solution?

Which of the following differential equations has y = x as one of its particular solution?

The general solution of the differential equation \[\frac{dy}{dx} = e^{x + y}\], is

A homogeneous differential equation of the form \[\frac{dx}{dy} = h\left( \frac{x}{y} \right)\] can be solved by making the substitution

Which of the following is a homogeneous differential equation?

The integrating factor of the differential equation \[x\frac{dy}{dx} - y = 2 x^2\]

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

The general solution of the differential equation \[\frac{y dx - x dy}{y} = 0\], is

The general solution of a differential equation of the type \[\frac{dx}{dy} + P_1 x = Q_1\] is

The general solution of the differential equation e^x dy + (y e^x + 2x) dx = 0 is

Determine the order and degree (if defined) of the following differential equation:-

\[\left( \frac{ds}{dt} \right)^4 + 3s\frac{d^2 s}{d t^2} = 0\]

Determine the order and degree (if defined) of the following differential equation:-

y"' + 2y" + y' = 0

Determine the order and degree (if defined) of the following differential equation:-

(y"')² + (y")³ + (y')⁴ + y⁵ = 0

Determine the order and degree (if defined) of the following differential equation:-

y"' + 2y" + y' = 0

Determine the order and degree (if defined) of the following differential equation:-

y" + (y')² + 2y = 0

Determine the order and degree (if defined) of the following differential equation:-

y" + 2y' + sin y = 0

Determine the order and degree (if defined) of the following differential equation:-

y"' + y² + e^y' = 0

Verify that the function y = e^−3x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + \frac{dy}{dx} - 6y = 0.\]

In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-

y = e^x + 1            y'' − y' = 0

In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-

y = x² + 2x + C            y' − 2x − 2 = 0

In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-

y = cos x + C            y' + sin x = 0

In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-

`y=sqrt(1+x^2)`                     `y'=(xy)/(1+x^2)`

In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-

y = x sin x              `xy'=y+xsqrt(x^2-y^2)`

In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-

`y=sqrt(a^2-x^2)`              `x+y(dy/dx)=0`

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

Form the differential equation representing the family of curves y = a sin (x + b), where a, b are arbitrary constant.

Form the differential equation representing the family of parabolas having vertex at origin and axis along positive direction of x-axis.

Form the differential equation of the family of circles having centre on y-axis and radius 3 unit.

Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.

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

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

Verify that xy = a e^x + b e^−x + x² is a solution of the differential equation \[x\frac{d^2 y}{d x^2} + 2\frac{dy}{dx} - xy + x^2 - 2 = 0.\]

Show that y = C x + 2C² is a solution of the differential equation \[2 \left( \frac{dy}{dx} \right)^2 + x\frac{dy}{dx} - y = 0.\]

Show that y² − x² − xy = a is a solution of the differential equation \[\left( x - 2y \right)\frac{dy}{dx} + 2x + y = 0.\]

Verify that y = A cos x + sin x satisfies the differential equation \[\cos x\frac{dy}{dx} + \left( \sin x \right)y=1.\]

Find the differential equation corresponding to y = ae^2x + be^−3x + ce^x where a, b, c are arbitrary constants.

Show that the differential equation of all parabolas which have their axes parallel to y-axis is \[\frac{d^3 y}{d x^3} = 0.\]

From x² + y² + 2ax + 2by + c = 0, derive a differential equation not containing a, b and c.

\[\frac{dy}{dx} = \sin^3 x \cos^4 x + x\sqrt{x + 1}\]

\[\frac{dy}{dx} = \frac{1}{x^2 + 4x + 5}\]

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

\[\frac{dy}{dx} + 4x = e^x\]

\[\frac{dy}{dx} = x^2 e^x\]

\[\frac{dy}{dx} - x \sin^2 x = \frac{1}{x \log x}\]

\[(\tan^2 x + 2\tan x + 5)\frac{dy}{dx} = 2(1+\tan x)\sec^2x\]

\[\frac{dy}{dx} = \sin^3 x \cos^2 x + x e^x\]

tan y dx + tan x dy = 0

(1 + x) y dx + (1 + y) x dy = 0

x cos² y dx = y cos² x dy

cos y log (sec x + tan x) dx = cos x log (sec y + tan y) dy

cosec x (log y) dy + x²y dx = 0

(1 − x²) dy + xy dx = xy² dx

\[\frac{dy}{dx} = \frac{\sin x + x \cos x}{y\left( 2 \log y + 1 \right)}\]

x (e^2y − 1) dy + (x² − 1) e^y dx = 0

\[\frac{dy}{dx} + 1 = e^{x + y}\]

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

cos (x + y) dy = dx

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

\[\frac{dy}{dx} = \frac{y\left( x - y \right)}{x\left( x + y \right)}\]

(x + y − 1) dy = (x + y) dx

\[\frac{dy}{dx} - y \cot x = cosec\ x\]

\[\frac{dy}{dx} - y \tan x = - 2 \sin x\]

\[\frac{dy}{dx} - y \tan x = e^x \sec x\]

\[\frac{dy}{dx} - y \tan x = e^x\]

(1 + y + x² y) dx + (x + x³) dy = 0

(x² + 1) dy + (2y − 1) dx = 0

`y sec^2 x + (y + 7) tan x(dy)/(dx)=0`

`(2ax+x^2)(dy)/(dx)=a^2+2ax`

(x³ − 2y³) dx + 3x² y dy = 0

x² dy + (x² − xy + y²) dx = 0

\[y - x\frac{dy}{dx} = b\left( 1 + x^2 \frac{dy}{dx} \right)\]

\[\frac{dy}{dx} + 2y = \sin 3x\]

\[\frac{dy}{dx} + y = 4x\]

\[\frac{dy}{dx} + 5y = \cos 4x\]

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

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

`x cos x(dy)/(dx)+y(x sin x + cos x)=1`

\[\left( 1 + y^2 \right) + \left( x - e^{- \tan^{- 1} y} \right)\frac{dy}{dx} = 0\]

\[y^2 + \left( x + \frac{1}{y} \right)\frac{dy}{dx} = 0\]

`2 cos x(dy)/(dx)+4y sin x = sin 2x," given that "y = 0" when "x = pi/3.`

(1 + y²) dx = (tan⁻¹ y − x) dy

`(dy)/(dx)+ y tan x = x^n cos x, n ne− 1`

Find the general solution of the differential equation \[\frac{dy}{dx} = \frac{x + 1}{2 - y}, y \neq 2\]

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

For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \frac{1 - \cos x}{1 + \cos x}\]

For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \sqrt{4 - y^2}, - 2 < y < 2\]

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

For the following differential equation, find the general solution:- `y log y dx − x dy = 0`

For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \sin^{- 1} x\]

For the following differential equation, find the general solution:- \[\frac{dy}{dx} + y = 1\]

For the following differential equation, find a particular solution satisfying the given condition:- \[x\left( x^2 - 1 \right)\frac{dy}{dx} = 1, y = 0\text{ when }x = 2\]

For the following differential equation, find a particular solution satisfying the given condition:- \[\cos\left( \frac{dy}{dx} \right) = a, y = 1\text{ when }x = 0\]

For the following differential equation, find a particular solution satisfying the given condition:- \[\frac{dy}{dx} = y \tan x, y = 1\text{ when }x = 0\]

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

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

Solve the following differential equation:- `y dx + x log  (y)/(x)dy-2x dy=0`

Solve the following differential equation:-

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

Solve the following differential equation:-

\[x\frac{dy}{dx} + 2y = x^2 , x \neq 0\]

Solve the following differential equation:-

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

Solve the following differential equation:-

\[\frac{dy}{dx} + 3y = e^{- 2x}\]

Solve the following differential equation:-

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

Solve the following differential equation:-

\[\frac{dy}{dx} + \left( \sec x \right) y = \tan x\]

Solve the following differential equation:-

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

Solve the following differential equation:-

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

Solve the following differential equation:-

(1 + x²) dy + 2xy dx = cot x dx

Solve the following differential equation:-

\[\left( x + y \right)\frac{dy}{dx} = 1\]

Solve the following differential equation:-

y dx + (x − y²) dy = 0

Solve the following differential equation:-

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

Find a particular solution of the following differential equation:- \[\left( 1 + x^2 \right)\frac{dy}{dx} + 2xy = \frac{1}{1 + x^2}; y = 0,\text{ when }x = 1\]

Find a particular solution of the following differential equation:- (x + y) dy + (x − y) dx = 0; y = 1 when x = 1

Find a particular solution of the following differential equation:- x² dy + (xy + y²) dx = 0; y = 1 when x = 1

Find the equation of the curve passing through the point (1, 1) whose differential equation is x dy = (2x² + 1) dx, x ≠ 0.

Find the equation of a curve passing through the point (−2, 3), given that the slope of the tangent to the curve at any point (x, y) is `(2x)/y^2.`

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

At any point (x, y) of a curve, the slope of the tangent is twice the slope of the line segment joining the point of contact to the point (– 4, –3). Find the equation of the curve given that it passes through (–2, 1).

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

Find the equation of a curve passing through the point (0, 1). If the slope of the tangent to the curve at any point (x, y) is equal to the sum of the x-coordinate and the product of the x-coordinate and y-coordinate of that point.

Find the equation of the curve passing through the origin given that the slope of the tangent to the curve at any point (x, y) is equal to the sum of the coordinates of the point.