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i have this 4th order ode question that i've been working on,

the homogeneous solution was easy enough by finding the particular solution has become a bit annoying,

the ode is

y'''' - 4y'' = 5x^{2}- e^{2x}

I have gotten the particular solution using variation of parameters, but I used mathematica to do the work, like finding the inverse of the wronskian matrix and integrating the ugly results.

the particular solution i get with mathematica is

y_{p}=1/192 (-30 - 3 E^(2 x) (-5 + 4 x) - 20 x^2 (3 + x^2))

I have checked that this is the particular solution by substitution into the ode and it gives required result.

My question is,

there is no way I was supposed to do this using mathematica, is there a way to approximate the particular solution, like with 2nd order odes?

I tried that in this case,

my particular solution I tried was

(A + Bx + Cx^{2})+De^{2x}

which is just the sum of the particular solution of x^2 and e^(2x)

Is there some way to estimate the particular solution without using variation of parameters in this question?

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# Homework Help: Particular solution to 4th order ode

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