Question

The solution of the differential equation $y\frac{dy}{dx}=x[\frac{y^2}{x^2}+\frac{\varphi \left(\frac{y^2}{x^2}\right)}{{\varphi }^{\prime }\left(\frac{y^2}{x^2}\right)}]$ is (where, $c$ is a constant)

• A. $\varphi \left(\frac{y^2}{x^2}\right)=cx$
• B. $x\varphi \left(\frac{y^2}{x^2}\right)=c$
• C. $\varphi \left(\frac{y^2}{x^2}\right)=c{x}^2$
• D. $x^2\varphi \left(\frac{y^2}{x^2}\right)=c$

Answer 1

The correct option is C $\varphi \left(\frac{y^2}{x^2}\right)=c{x}^2$

Given differential equation can be written as
$\frac{dy}{dx}=\frac{y}{x}+\frac{x\varphi \left(\frac{y^2}{x^2}\right)}{y{\varphi }^{\prime }\left(\frac{y^2}{x^2}\right)}$

Put $y=vx\Rightarrow \frac{dy}{dx}=v+x\frac{dv}{dx}$
$\therefore$ Given equation becomes,
$v+x\frac{dv}{dx}=\frac{vx}{x}+\frac{x\varphi \left(\frac{v^2{x}^2}{x^2}\right)}{vx{\varphi }^{\prime }\left(\frac{v^2{x}^2}{x^2}\right)}$

$\Rightarrow x\frac{dv}{dx}=\frac{\varphi \left(v^2\right)}{v{\varphi }^{\prime }\left(v^2\right)}$

$\Rightarrow \frac{v{\varphi }^{\prime }\left(v^2\right)}{\varphi \left(v^2\right)}dv=\frac{dx}{x}$

On integrating both sides, we get
$\frac{1}{2}\log \varphi \left(v^2\right)=\log x+\log c_1$

$\Rightarrow \log \varphi \left(v^2\right)=2\log x{c}_1$

$\Rightarrow \varphi \left(v^2\right)={\left(x{c}_1\right)}^2\Rightarrow \varphi \left(\frac{y^2}{x^2}\right)=x^2{c}_1^2$

$\Rightarrow \varphi \left(\frac{y^2}{x^2}\right)=x^{2}c$ [put $c_1^2=c$]