51
MCQ (Single Correct Answer)
The solution of $\frac{d y}{d x}+y=e^{-x}, y(0)=0$ is
A
$y=e^x(x-1)$
B
$y=x e^{-x}$
C
$y=x e^{-x}+1$
D
$y=(x+1) \mathrm{e}^{-x}$
52
MCQ (Single Correct Answer)
The integrating factor of differential equation $\frac{d y}{d x}+y \tan x-\sec x=0$ is
A
$\cos x$
B
$\sec x$
C
$e^{\cos x}$
D
$\mathrm{e}^{\sec x}$
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
$\frac{d^2 y}{d x^2}+m^2 y=0$