Question

For differential equation ${\left(\frac{dy}{dx}\right)}^2-x\left(\frac{dy}{dx}\right)+y=0$,find the solution

Answer 1

${\left(\frac{dy}{dx}\right)}^2-x\left(\frac{dy}{dx}\right)+y=0$

Let $\frac{dy}{dx}=p$ so we get

$p^2-xp+y=0$

Differentiating w.r.t ${}^{\prime }{x}^{\prime }$

$2p\frac{dp}{dx}-p-x.\frac{dp}{dx}+\frac{dy}{dx}=0$

$2p\frac{dp}{dx}-x.\frac{dp}{dx}-p+p=0$

$2p.\frac{dp}{dx}-x\frac{dp}{dx}=0$

So $2p-x=0\Rightarrow 2p=x$

As $\frac{dy}{dx}=p$ So $\frac{dy}{dx}=\frac{x}{2}$

Integrating both sides

$\int dy=\int \frac{x}{2}.dx$

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