CBSE - Class 12 Mathematics Differential Equations Worksheet
1.
Integrating factor of $x\frac{dy}{dx} – y = x^4 – 3x$ is :
a.
$x$
b.
$\log x$
c.
$\frac{1}{x}$
d.
$-x$
2.
Given that $\frac{dy}{dx} = e^{-2y}$ and $y = 0$ when $x = 5$. Find the value of $x$ when $y = 3$.
3.
Find the equation of a curve passing through origin if the slope of the tangent to the curve at any point $(x, y)$ is equal to the square of the difference of the abcissa and ordinate of the point.
4.
Find the equation of a curve passing through origin and satisfying the differential equation $(1+x^2)\frac{dy}{dx} + 2xy = 4x^2$.
5.
The integrating factor of the differential equation $\frac{dy}{dx} + (1+\frac{1}{x})y = ...$ is:
a.
$\frac{x}{e^x}$
b.
$\frac{xe^x}{x}$
c.
$x e^x$
d.
$e^x$
6.
Fill in the blanks: $\frac{dy}{dx} + \frac{y}{x\log x} = \frac{1}{x}$ is an equation of the type _________.
7.
The differential equation of the family of curves $x^2 + y^2 – 2ay = 0$, where $a$ is arbitrary constant, is:
a.
$(x^2 – y^2)\frac{dy}{dx} = 2xy$
b.
$2 (x^2 + y^2)\frac{dy}{dx} = xy$
c.
$2 (x^2 – y^2)\frac{dy}{dx} = xy$
d.
$(x^2 + y^2)\frac{dy}{dx} = 2xy$
8.
Solution of $\frac{dy}{dx} - y = 1$, $y (0) = 1$ is given by
a.
$xy = – e^x$
b.
$xy = – e^{-x}$
c.
$xy = – 1$
d.
$y = 2 e^x – 1$
9.
The order and degree of the differential equation $[1 + (\frac{dy}{dx})^2]^{\frac{3}{2}} = \frac{d^2y}{dx^2}$ are :
a.
2, $\frac{3}{2}$
b.
2, 3
c.
2, 1
d.
3, 4
10.
Solve : $x\frac{dy}{dx} = y (\log y – \log x + 1)$
11.
State True or False: The solution of $\frac{dy}{dx} = (\frac{y}{x})^{\frac{1}{3}}$ is $y^{\frac{2}{3}} – x^{\frac{2}{3}} = c$.
12.
State True or False: Differential equation representing the family of curves $y = e^x (A\cos x + B\sin x)$ is $\frac{d^2y}{dx^2} – 2\frac{dy}{dx} + 2y = 0$.
13.
Solve : $(x + y) (dx – dy) = dx + dy$.[Hint: Substitute $x + y = z$ after seperating $dx$ and $dy$]
14.
Solve the differential equation $\frac{dy}{dx} = 1 + x + y^2 + xy^2$, when $y = 0$, $x = 0$.
