For the differential equation $y^{\prime\prime}+ 36 y = \cos\!\left(6x\right) - 6 x^2$

Part 1: Solve the homogeneous equation
The differential operator for the homogeneous equation is
List the complementary functions

Part 2: Find the particular solution
To solve the non-homogeneous differential equation, we look for functions annihilated by the differential operator (a multiple of the differential operator from above)
Therefore the particular solution must be made up of the functions
Substituting these into the differential equation, we find the particular solution is

Part 3: Solve the non-homogeneous equation
$y^{\prime\prime}+ 36 y = \cos\!\left(6x\right) - 6 x^2$ has general solution (remember to use the format I gave you in your correct answer to the complementary functions above)

Now that we have the general solution solve the IVP
$y(0)=-6$
$y^\prime(0)=5$


Here is a graph of the solution to the IVP