differential equations

Question

Consider the ODE y/x y' + y(y − 1) = 0.

(a) Verify that y1 = e^(− 1/2 x^2) is a solution to the ODE y/x y'+ y^2 = 0.

(b) Use the variation of parameters substitution y = µy1 to find a family of solutions to the ODE.

Answer 1

Take first derivative dy/dx, same as y', plug y'all in the ODE to verify the answer;

y' = e ^(-1/2 x²) x (-1/2 * 2x ) = -xy after simplification.

Now plug y' in ODE, WE get:

(y/x) * (-xy) + y²

^{this Is = 0 , hence proved.}