How to solve this ODE ?$ (x^2) y”” + (3x^2-2x)y”’ + (3x^2-4x+2)y” +(x^2-2x+2)y’ = 0 $ ( hint )


$ (x^2) y”” + (3x^2-2x)y”’ + (3x^2-4x+2)y” +(x^2-2x+2)y’ = 0 $

i tried to Guess a solution and use the fact that i can decrease the ODE to less power ( to $ y”’$ ) by using the Wronskian .

i Guessed that $ e^{-x} $ is a solution . is it the way we solve this kind of equations ? its homework question so i guess i can solve it.

euler ODE doesn’t work here .

Solving for $y” – 4y’ – 5y – 2 = 0$

I am looking to solve for the above nonhomogeneous ODE. I know how to find the general solution for the reduced equation of the homogeneous form, that is, $ $ y” – 4y’ – 5y = 0.$ $

The characteristic equation is $ r^{2} – 4r – 5 = 0$ , which gives two real and distinct roots $ r=-1,5$ .

So the complementary solution is $ y_{c}=c_{1}e^{5x} + c_{2}e^{-x}$ .

Now I am looking to guess the particular on the right-hand side but I am not sure about how to do that in order to find the general solution of the above nonhomogenous ODE.