Question

Find the particular solution of the differential equation $\frac{dy}{dx}+2y\cot x=4xcosecx(x\ne 0)$, given that $y=0$, when $x=\frac{\pi }{2}$

Answer 1

$\frac{dy}{dx}+Py=Q$ has the integrating factor as $e^{\int Pdx}$.

Here, $P=2\cot x,Q=4xcosecx$

Integrating factor $=e^{2\int \cot xdx}=e^{2\log \left(\sin x\right)}={\sin }^{2}x$

$\Rightarrow \frac{d\left(y{\sin }^{2}x\right)}{dx}=4x\times cosecx\times {\sin }^{2}x$

$\Rightarrow d\left(y{\sin }^{2}x\right)=4x\sin xdx$

Integrating both sides, we have

$y{\sin }^{2}x=4x(-\cos x)+4\sin x+c$

Substituting $y=0$ when $x=\frac{\pi }{2}$, we get

$0=0+4+c$

$\Rightarrow c=-4$

The solution thus becomes $y{\sin }^{2}x=4(-x\cos x+\sin x-1)$