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

${y}^{\prime}=\frac{z-{z}^{\prime}x}{{z}^{2}}$

$\frac{z-{z}^{\prime}x}{{z}^{2}}-\frac{1}{2z}=x\mathrm{sin}(z)$

multiply by ${z}^{2}$

$z-{z}^{\prime}x-\frac{z}{2}={z}^{2}x\mathrm{sin}(z)$

And now I have no idea how to manipulate this.

${y}^{\prime}\left(x\right)-\frac{y\left(x\right)}{2x}=x\mathrm{sin}\left(\frac{x}{y\left(x\right)}\right)$

I think it is separable variable differential equations. I tried to substitute:

$z=\frac{x}{y}$

and

${y}^{\prime}=\frac{z-{z}^{\prime}x}{{z}^{2}}$

$\frac{z-{z}^{\prime}x}{{z}^{2}}-\frac{1}{2z}=x\mathrm{sin}(z)$

multiply by ${z}^{2}$

$z-{z}^{\prime}x-\frac{z}{2}={z}^{2}x\mathrm{sin}(z)$

And now I have no idea how to manipulate this.

Hadley Cunningham

Beginner2022-06-22Added 20 answers

Hint

Let $t={\displaystyle \frac{y}{x}}$,

Then $y=xt$

$\frac{dy}{dx}}=t+x{\displaystyle \frac{dt}{dx}$

$\therefore t+x{\displaystyle \frac{dt}{dx}}-{\displaystyle \frac{t}{2}}=x\mathrm{sin}{\displaystyle \frac{1}{t}}$

$x{\displaystyle \frac{dt}{dx}}=x\mathrm{sin}{\displaystyle \frac{1}{t}}-{\displaystyle \frac{t}{2}}$

$(x\mathrm{sin}{\displaystyle \frac{1}{t}}-{\displaystyle \frac{t}{2}}){\displaystyle \frac{dx}{dt}}=x$

This belongs to an Abel equation of the second kind.

Let $t={\displaystyle \frac{y}{x}}$,

Then $y=xt$

$\frac{dy}{dx}}=t+x{\displaystyle \frac{dt}{dx}$

$\therefore t+x{\displaystyle \frac{dt}{dx}}-{\displaystyle \frac{t}{2}}=x\mathrm{sin}{\displaystyle \frac{1}{t}}$

$x{\displaystyle \frac{dt}{dx}}=x\mathrm{sin}{\displaystyle \frac{1}{t}}-{\displaystyle \frac{t}{2}}$

$(x\mathrm{sin}{\displaystyle \frac{1}{t}}-{\displaystyle \frac{t}{2}}){\displaystyle \frac{dx}{dt}}=x$

This belongs to an Abel equation of the second kind.