Question

Solution of the differential equation $\frac{dy}{dx}+\frac{y}{x}=\sin x$ is:

• A. $x(y+\cos x)=\sin x+c$
• B. $x(y-\cos x)=\sin x+c$
• C. $x(y\cos x)=\sin x+c$
• D. $x(y-\cos x)=\cos x+c$
• E. $x(y+\cos x)=\cos x+c$

Answer 1

Answer: (A)

$\because$ $\frac{dy}{dx}+\frac{y}{x}=\sin x$
Here, $P=\frac{1}{x}$ and $Q=\sin x$
$\therefore$ $IF=e^{\int pdx}=e^{\int 1/xdx}=x$
$\therefore$ Solution is $y(IF)=\int Q(IF)dx+c$
$\Rightarrow$ $y.x=\int x\sin xdx+c$
$\Rightarrow$ $xy=-x\cos x+\sin x+c$
$\Rightarrow$ $x(y+\cos x)=\sin x+c$