Use an appropriate change of variables to solve the differential equation. Use an appropriate change of variables to solve the differential equation.
$$t\frac{dy}{dt}-y=\sqrt{t^2+y^2}$$
My friend and I are trying to figure out how to solve this equation. Our professor has given us several methods but we aren't sure which to use because none of the equations are similar to this one.
Any help would be appreciated especially if you could help us with step by step.
Thanks!
--
UPDATE:
$$t\frac{dy}{dt}-y=\sqrt{t^2+y^2}$$
$$\frac{dy}{dt}-\frac{y}t=\sqrt{1+\frac{y^2}{t}}$$ where u= y/t
$$\frac{dy}{dt}=\sqrt{1+u^2}+u=f(u)$$
$$f(u)-u=\sqrt{1+u^2}+u-u=\sqrt{1+u^2}$$
...
We ended up with $ln|\sqrt{1+u^2}|=ln|x|+c$

HINT
One easy one to try is $y=zt$ so $y'=z't+z$ and the ODE becomes $$z't = \sqrt{1+z^2}$$ which is separable.
