The set F of all functions from R to R is a vector space with the usual operations of addition of functions and scalar multiplication. Is the set of solutions to the differential equation f''(x)+3f'(x)+(x^2)f(x)=0 a subspace of F? Justify your answer I know that to prove that the set of solutions is a subspace of F I need to show that the set not empty, is closed under addition and closed under scalar multiplication. The only problem I have is solving the differential equation which i am not sure how to do because solving this kind of differential equation (I only know how to solve second order DE's with constant coefficients) has not brought up in the current course (Linear mathematics 2 year maths) or any of the prerequisite coursess . Do I actually need to solve the equation to find the answer or is their another way to find if its a subspace of F or not? Any help would be very much appreciated.