Hey I need to learn how to solve a problem using Partial fraction expansion theory

the problem is:

Thank you!Quote:

y''+y=x^2+e^(2x)

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- Dec 5th 2010, 06:58 PMdeltafeePartial Fraction expansion
Hey I need to learn how to solve a problem using Partial fraction expansion theory

the problem is:

Quote:

y''+y=x^2+e^(2x)

- Dec 5th 2010, 08:39 PMProve It
Can't you just solve this using inspired guesswork?

.

First evaluating the homogeneous solution, the characteristic equation is

.

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

.

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

.