53
MCQ (Single Correct Answer)
The solution of differential equation $\frac{d y}{d x}=\frac{1+y^2}{1+x^2}$ is
A
$y=\tan ^{-1} x$
B
$y-x=k(1+x y)$
C
$x=\tan ^{-1} y$
D
$\tan (x y)=k$
54
MCQ (Single Correct Answer)
The integrating factor of differential equation $\frac{d y}{d x}+y=\frac{1+y}{x}$ is
A
$\frac{x}{\mathrm{e}^x}$
B
$\frac{\mathrm{e}^x}{x}$
C
$x \mathrm{e}^x$
D
$\mathrm{e^x}$
55
MCQ (Single Correct Answer)
$y=a e^{m x}+b e^{-m x}$ satisfies which of the following differential equation?
A
$\frac{d y}{d x}+m y=0$
B
$\frac{d y}{d x}-m y=0$
C
$\frac{d^2 y}{d x^2}-m^2 y=0$
D
$\frac{d^2 y}{d x^2}+m^2 y=0$
56
MCQ (Single Correct Answer)
The solution of differential equation $\cos x \sin y d x+\sin x \cos y d y=0$ is
A
$\frac{\sin x}{\sin y}=C$
B
$\sin x \sin y=C$
C
$\sin x+\sin y=C$
D
$\cos x \cos y=C$
57
MCQ (Single Correct Answer)
The solution of $x \frac{d y}{d x}+y=e^x$ is
A
$y=\frac{\mathrm{e}^x}{x}+\frac{k}{x}$
B
$y=x e^x+C x$
C
$y=x e^x+k$
D
$x=\frac{e^y}{y}+\frac{k}{y}$