Question

A solution of the differential equation ${\left(\frac{dy}{dx}\right)}^2-x\frac{dy}{dx}+y=0$ is

• A. y =2
• B. y =2x
• C. y =2x -4
• D. $y=2x^2-4$

Answer 1

The correct option is C y =2x -4

Given equation can be written as $y=x\frac{dy}{dx}-{\left(\frac{dy}{dx}\right)}^2$
If $\frac{dy}{dx}=p$, then $y=px-p^2$
Differentiating w.r.t. x, we get
$p=p+x\frac{dp}{dx}-2p\frac{dp}{dx}\Rightarrow \frac{dp}{dx}(x-2p)=0\Rightarrow \frac{dp}{dx}=0$
Integrating w.r.t. x, we get p =c
$\therefore \frac{dy}{dx}=c;\therefore y=cx-c^2$
If c =2, then y =2x -4.