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  4. The solution of differential equation y y'=x((y2/x2)+(f(y2 / x2)/f'(y2 / x2))) is

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

Differential Equations

Solution:

The given equation can be written as
$\frac{y}{x} \frac{d y}{d x}=\left\{\frac{y^{2}}{x^{2}}+\frac{f\left(y^{2} / x^{2}\right)}{f'\left(y^{2} / x^{2}\right)}\right\}$
The above equation is a homogeneous equation.
Putting $y$ $=v x$, we get
$v\left[v+x \frac{d v}{d x}\right]=v^{2}+\frac{f\left(v^{2}\right)}{f'\left(v^{2}\right)}$
or $v x \frac{d v}{d x}=\frac{f\left(v^{2}\right)}{f'\left(v^{2}\right)}$ or
$\frac{2 v f'\left(v^{2}\right)}{f\left(v^{2}\right)} d v=2 \frac{d x}{x}$
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 $f\left(y^{2} / x^{2}\right)=c x^{2}$