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 $y=2x-4$

$y=-{\left(\frac{dy}{dx}\right)}^2+x\frac{dy}{dx}\Rightarrow \frac{dy}{dx}=x\frac{d^{2}y}{{dx}^2}+\frac{dy}{dx}-2\frac{dy}{dx}\frac{d^{2}y}{{dx}^2}\Rightarrow \frac{dy}{dx}=\frac{dy}{dx}+\frac{d^{2}y}{{dx}^2}(x-2\frac{dy}{dx})\Rightarrow \frac{d^{2}y}{{dx}^2}(x-2\frac{dy}{dx})=0$
$\Rightarrow \frac{d^{2}y}{{dx}^2}=0$ or $x-2\frac{dy}{dx}=0$
$\Rightarrow y=cx-c^2$ or $y=\frac{x^2}{4}$