Solve \frac{dy}{dx}=\sin{\frac{y}{x}} | ODR Problem

[Vrbic]

Hello,
I have a problem with solution this equation: $\frac{dy}{dx}=\sin{\frac{y}{x}}$. I tried to employ substitution $u=\frac{y}{x}$. From this comming $x\frac{du}{dx}=\sin{u}-u$. But than integration $\int=\frac{1}{\sin{u}-u}$ and I don't know if it is possible to solve it or all this way is wrong.
Thank you for comment or advice.

 

[gtoguy97]

Unfortunately, this equation cannot be solved in terms of elementary functions. However, there are various numerical methods available which can be used to obtain approximate solutions. For example, using the Euler's method, you can discretize the differential equation as follows:\frac{y(x+h)-y(x)}{h} = \sin \left(\frac{y(x)}{x}\right),where h is a small step size. This equation can then be solved iteratively for different values of x and h to obtain approximate solutions. There are also more sophisticated numerical methods like Runge-Kutta which can be used to obtain better approximations.