Hey I need to learn how to solve a problem using Partial fraction expansion theory
the problem is:
Thank you!y''+y=x^2+e^(2x)
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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