Question

Find a particular solution of the differential equation $\frac{dy}{dx}+y\cot x=4xcosecx$.

Answer 1

Given

$\frac{dy}{dx}+y\cot x=4xcosecx$

This of the form

$\frac{dy}{dx}+py=\theta$

where $p=\cot x$ & $\theta =4xcosecx$

If $=e^{\int pdx}=e^{\int \cot xdx}$

If $=e^{log\left(\sin x\right)}$

If $=\sin x$

(As $e^{logx}=x$)

solution is

$x\left(if\right)=\int (\theta \times I-F-)dx+c$

$x\sin x=\int 4xcosecx\sin xdx+c$

$x\sin x=\int 4x\times \frac{1}{\sin x}\times \sin xdx+c$

$y\sin x=\int 4xdx+c$

$y\sin x=\frac{4x^2}{2}+c$

$y\sin x=2x^2+c$.