Question

The solution of the differential equation $\frac{dy}{dx}=\frac{y}{x}+\frac{\varphi \left(\frac{y}{x}\right)}{\varphi \left(\frac{y}{x}\right)}$ is

• A. $\varphi \left(\frac{y}{x}\right)=kx$
  
• B. $x\varphi \left(\frac{y}{x}\right)=k$
  
• C. $\varphi \left(\frac{y}{x}\right)=ky$
  
• D. $y\varphi \left(\frac{y}{x}\right)=k$

Answer 1

The correct option is A $\varphi \left(\frac{y}{x}\right)=kx$


$\frac{dy}{dx}=\frac{y}{x}+\frac{\varphi \left(\frac{y}{x}\right)}{{\varphi }^{\prime }\left(\frac{y}{x}\right)}$. Put $y=vx\Rightarrow \frac{dy}{dx}=v+x\frac{dv}{dx}$
$\therefore$ The given differential equation becomes
$v+x\frac{dv}{dx}=v+\frac{\varphi \left(v\right)}{{\varphi }^{\prime }\left(v\right)}\Rightarrow \frac{{\varphi }^{\prime }\left(v\right)}{\varphi \left(v\right)}dv=\frac{dx}{x}\Rightarrow log\varphi \left(v\right)=logx+logk\Rightarrow \varphi \left(v\right)=kx\Rightarrow \varphi \left(\frac{y}{x}\right)=kx$