Consider the following, Linear (but NOT Constant Coefficient), 2nd Order ODE: x²y" -x(x+2)y'+(x+2) y = f(x) x>0
a) Verify that y=x is a solution to the homogeneous problem: f(x) = 0.
b) Use reduction of order to show that a second homogeneous solution is y = xe^x.
c) Use the Wronskian to prove that these two solutions are linearly independent. (For all x?)
d) Use Variation of Parameters to find the general solution to the non-homogeneous problem when f(x)= 2x³. (Hint: Remember the form of the ODE used to derive the Variation of Parameters formula!)