Question

The solution of differential equation $y{y}^{\prime }=x\left(\frac{y^2}{x^2}+\frac{f\left(y^2/x^2\right)}{f^{\prime }\left(y^2/x^2\right)}\right)$ is

• A. $f\left(y^2/x^2\right)=c{x}^2$
• B. $x^{2}f\left(y^2/x^2\right)=c^2{y}^2$
• C. $x^{2}f\left(y^2/x^2\right)=c$
• D. $f\left(y^2/x^2\right)=cy/x$

Answer 1

Answer: (A)

The given equation can be written as
$\frac{y}{x}\frac{dy}{dx}=\left\{\frac{y^2}{x^2}+\frac{f\left(y^2/x^2\right)}{f^{\prime }\left(y^2/x^2\right)}\right\}$
The above equation is a homogeneous equation.
Putting $y$ $=vx$, we get
$v\left[v+x\frac{dv}{dx}\right]=v^2+\frac{f\left(v^2\right)}{f^{\prime }\left(v^2\right)}$
or $vx\frac{dv}{dx}=\frac{f\left(v^2\right)}{f^{\prime }\left(v^2\right)}$ or
$\frac{2v{f}^{\prime }\left(v^2\right)}{f\left(v^2\right)}dv=2\frac{dx}{x}$
Now, integrating both sides, we get
$\log f\left(v^2\right)=\log x^2+\log c$
or $\log f\left(v^2\right)=\log c{x}^2$ or
$f\left(v^2\right)=c{x}^{2}c=$ constant $]$
or $f\left(y^2/x^2\right)=c{x}^2$