Llubanipo

2022-06-21

I am trying to solve the differential equation
${y}^{\prime }\left(x\right)-\frac{y\left(x\right)}{2x}=x\mathrm{sin}\left(\frac{x}{y\left(x\right)}\right)$
I think it is separable variable differential equations. I tried to substitute:
$z=\frac{x}{y}$
and
${y}^{\prime }=\frac{z-{z}^{\prime }x}{{z}^{2}}$
$\frac{z-{z}^{\prime }x}{{z}^{2}}-\frac{1}{2z}=x\mathrm{sin}\left(z\right)$
multiply by ${z}^{2}$
$z-{z}^{\prime }x-\frac{z}{2}={z}^{2}x\mathrm{sin}\left(z\right)$
And now I have no idea how to manipulate this.

### Answer & Explanation

Hint
Let $t=\frac{y}{x}$,
Then $y=xt$
$\frac{dy}{dx}=t+x\frac{dt}{dx}$
$\therefore t+x\frac{dt}{dx}-\frac{t}{2}=x\mathrm{sin}\frac{1}{t}$
$x\frac{dt}{dx}=x\mathrm{sin}\frac{1}{t}-\frac{t}{2}$
$\left(x\mathrm{sin}\frac{1}{t}-\frac{t}{2}\right)\frac{dx}{dt}=x$
This belongs to an Abel equation of the second kind.

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