Can't you just solve this using inspired guesswork?

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First evaluating the homogeneous solution, the characteristic equation is

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Since these solutions are complex conjugates of the form , the solution is of the form . In this case and .

So the homogeneous solution is .

Now to look for a particular solution, since the RHS of your DE is , a particular solution might be .

That means

and .

So substituting into the DE gives

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Equating like coefficients gives .

Therefore .

So your particular solution is .

Adding your homogeneous solution to your particular solution gives the general solution, so the general solution is

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