Family $y = Ax + A^3$ of curves is represented by the differential equation of degree :
a.
1
b.
2
c.
3
d.
4
12.
State True or False: The solution of $\frac{dy}{dx} = (\frac{y}{x})^{\frac{1}{3}}$ is $y^{\frac{2}{3}} – x^{\frac{2}{3}} = c$.
13.
Find the equation of a curve passing through origin and satisfying the differential equation $(1+x^2)\frac{dy}{dx} + 2xy = 4x^2$.
14.
The solution of the differential equation $\frac{dy}{dx} = \frac{1+y^2}{1+x^2}$ is:
a.
$y = \tan^{-1}x$
b.
$y – x = k (1 + xy)$
c.
$x = \tan^{-1}y$
d.
$\tan (xy) = k$
15.
Fill in the blanks: The solution of $(1 + x^2) \frac{dy}{dx} +2xy – 4x^2 = 0$ is _________.
16.
State True or False: Differential equation representing the family of curves $y = e^x (A\cos x + B\sin x)$ is $\frac{d^2y}{dx^2} – 2\frac{dy}{dx} + 2y = 0$.
17.
Integrating factor of the differential equation $(1 – x^2)\frac{dy}{dx} - xy = 1$ is
a.
$-x$
b.
$\frac{x}{1+x^2}$
c.
$\sqrt{1-x^2}$
d.
$\frac{1}{2} \log (1 – x^2)$
18.
State True or False: Correct substitution for the solution of the differential equation of the type $\frac{dx}{dy} = g(x, y)$ where $g(x, y)$ is a homogeneous function of the degree zero is $x = vy$.
19.
Solve the differential equation $dy = \cos x (2 – y \csc x) dx$ given that $y = 2$ when $x = \frac{\pi}{2}$.
20.
The order and degree of the differential equation $(\frac{d^2y}{dx^2})^{\frac{1}{2}} + (\frac{dy}{dx})^{\frac{1}{5}} + 4 = 0$, respectively, are
a.
2 and not defined
b.
2 and 2
c.
2 and 3
d.
3 and 3
