Question

Find the particular solution of the differential equation:
$x\frac{dy}{dx}-y+x\sin \left(\frac{y}{x}\right)=0$, given that when $x=2,y=\pi .$

Answer 1

$x\frac{dy}{dx}-y+xsin\left(\frac{y}{x}\right)=0$

Dividing throughout by $x$, we have

$\frac{dy}{dx}-\frac{y}{x}+sin\left(\frac{y}{x}\right)=0$

Let $\frac{y}{x}=t,\therefore dy=tdx+xdt$

Substituting these, we can write

$t+x\frac{dt}{dx}-t-sint=0$

$\Rightarrow \frac{dt}{sint}=\frac{dx}{x}$

$\Rightarrow \int cosectdt=\int \frac{dx}{x}$

$\int cosect\left(\frac{cosect+cott}{coesct+cott}\right)dt=\ln x+c$

$\therefore -\ln (cosect+cott)=\ln x+c$

$\Rightarrow -c=\ln (xcosec\left(\frac{y}{x}\right)+xcot\left(\frac{y}{x}\right))$

Substituting when $x=2,y=\pi$, we have

$-c=\ln (2\times 1+0)=\ln 2$

Thus, we can write the solution as

$\Rightarrow 1=2(xcosec\left(\frac{y}{x}\right)+xcot\left(\frac{y}{x}\right))$