44
MCQ (Single Correct Answer)
The solution of $\frac{d y}{d x}-y=1, y(0)=1$ is given by
A
$x y=-\mathrm{e}^x$
B
$x y=-\mathrm{e}^{-x}$
C
$x y=-1$
D
$y=2 e^x-1$
45
MCQ (Single Correct Answer)
The number of solutions of $\frac{d y}{d x}=\frac{y+1}{x-1}$, when $y(1)=2$ is
A
none
B
one
C
two
D
infinite
46
MCQ (Single Correct Answer)
Which of the following is a second order differential equation?
A
$\left(y^{\prime}\right)^2+x=y^2$
B
$y^{\prime} y^{\prime \prime}+y=\sin x$
C
$y^{\prime \prime \prime}+\left(y^{\prime \prime}\right)^2+y=0$
D
$y^{\prime}=y^2$
47
MCQ (Single Correct Answer)
The integrating factor of differential equation $\left(1-x^2\right) \frac{d y}{d x}-x y=1$ is
A
$-x$
B
$\frac{x}{1+x^2}$
C
$\sqrt{1-x^2}$
D
$\frac{1}{2} \log \left(1-x^2\right)$
48
MCQ (Single Correct Answer)
$\tan ^{-1} x+\tan ^{-1} y=C$ is general solution of the differential equation
A
$\frac{d y}{d x}=\frac{1+y^2}{1+x^2}$
B
$\frac{d y}{d x}=\frac{1+x^2}{1+y^2}$
C
$\left(1+x^2\right) d y+\left(1+y^2\right) d x=0$
D
$\left(1+x^2\right) d x+\left(1+y^2\right) d y=0$