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CBSE - Class 12 Mathematics Differential Equations Worksheet

Fill in the blanks: The integrating factor of $\frac{dy}{dx} + \frac{y}{x(y+1)} = 1$ is _________.

Find the equation of the curve through the point (1, 0) if the slope of the tangent to the curve at any point $(x, y)$ is $\frac{y-1}{x^2+x}$.

Find the general solution of $\frac{dy}{dx} + ay = e^{mx}$.

State True or False: The solution of $\frac{dy}{dx} = (\frac{y}{x})^{\frac{1}{3}}$ is $y^{\frac{2}{3}} – x^{\frac{2}{3}} = c$.

Fill in the blanks: The solution of the differential equation $x\frac{dy}{dx} + 2y = x^2$ is _________.

Solution of the differential equation $\tan y \sec^2x dx + \tan x \sec^2y dy = 0$ is :

a. $\tan x + \tan y = k$ b. $\tan x – \tan y = k$ c. $\frac{\tan x}{\tan y} = k$ d. $\tan x . \tan y = k$

Find the differential equation of system of concentric circles with centre (1, 2).

The solution of the equation $(2y – 1) dx – (2x + 3)dy = 0$ is :

a. $\frac{2x-1}{2y+3} = k$ b. $\frac{2y+1}{2x-3} = k$ c. $\frac{2x+3}{2y-1} = k$ d. $\frac{2x-1}{2y-1} = k$

Which of the following is a second order differential equation?

a. $(y?)^2 + x = y^2$ b. $y?y? + y = \sin x$ c. $y?? + (y?)^2 + y = 0$ d. $y? = y^2$

10.

If $y = e^{-x} (A\cos x + B\sin x)$, then $y$ is a solution of

a. $\frac{d^2y}{dx^2} + 2\frac{dy}{dx} = 0$ b. $\frac{d^2y}{dx^2} - 2\frac{dy}{dx} + 2y = 0$ c. $\frac{d^2y}{dx^2} + 2\frac{dy}{dx} + 2y = 0$ d. $\frac{d^2y}{dx^2} + 2y = 0$

11.

Integrating factor of the differential equation $\frac{dy}{dx} + y \tan x - \sec x = 0$ is:

a. $\cos x$ b. $\sec x$ c. $e^{\cos x}$ d. $e^{\sec x}$

12.

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

13.

Find the general solution of the differential equation $(1 + y^2) + (x – e^{\tan^{-1}y})\frac{dy}{dx} = 0$.

14.

Solve : $\frac{d}{dx}(y + xy) = x (\sin x + \log x)$

15.

The differential equation for which $y = a\cos x + b\sin x$ is a solution, is :

a. $\frac{d^2y}{dx^2} + y = 0$ b. $\frac{d^2y}{dx^2} – y = 0$ c. $\frac{d^2y}{dx^2} + (a + b) y = 0$ d. $\frac{d^2y}{dx^2} + (a – b) y = 0$

16.

Fill in the blanks: The degree of the differential equation $\sqrt{1 + (\frac{dy}{dx})^2} = x$ is _________.

17.

State True or False: Integrating factor of the differential of the form $\frac{dx}{dy} + p_1 x = Q_1$ is given by $e^{\int p_1 dy}$.

18.

Given that $\frac{dy}{dx} = e^{-2y}$ and $y = 0$ when $x = 5$. Find the value of $x$ when $y = 3$.

19.

Which of the following is the general solution of $\frac{d^2y}{dx^2} - 2\frac{dy}{dx} + 2y = 0$ ?

a. $y = (Ax + B)e^x$ b. $y = (Ax + B)e^{-x}$ c. $y = Ae^x + Be^{-x}$ d. $y = A\cos x + B\sin x$

20.

The general solution of the differential equation $\frac{dy}{dx} = e^{x^2} + 2xy$ is :

a. $y = ce^{-x^2}$ b. $y = ce^{x^2}$ c. $y = (x+c)e^{x^2}$ d. $y = (c-x)e^{x^2}$

Worksheet Answers

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