Hello guys any help would be appreciated for this question.

Show that the general solution of y=x*p + f(p) is y=cx+f(c), where p=dy/dx and c is an arbitrary constant. Also show that there is a 2nd solution obeying the diff. equation d/dp(f(p)) + x =0. Finally find the singular solution obeying both of the diff equations.

Obviously the first equation is not a separable equation. Also can't use the integrating factor since we have a function of the derivative in our equation. Can anyone tell me how do we solve this type of differential equations since is the first time I see this type? By direct differentiation of the given solution is easy to show that it is a solution but I don't think so this is a way. And after solving the first equation how do I proceed for the other parts?

Thanks in advance for any help!