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I'm stuck on this one and it's due tomorrow so I'd love some help.
Solve y''-y'+y=x+e^x
Another approach would be the annihilator method.
We are given:
(1)
Observe thatannihilates
and
annihilates
and so define:
Applyingto both sides of the ODE we get:
(2)
Thus, the auxiliary equation is:
The roots are:
Thus, a general solution to (1) is:
(3)
Now, recall that a general solution to (1) is of the form. Since every solution to (1) is also a solution to (2), then
must have the form displayed on the right-hand side of (3). But, we recognize that:
and so there must exist a particular solution of the form:
In order to substitute this into (1), we must first compute:
and so we find:
Equating coefficients, we find:
and so:
and we have the solution:
=1+x+e^x+c_1e^{ \frac{x}{2}}\cos\left( \frac{ \sqrt{3}}{2}x \right)+c_2e^{ \frac{x}{2}}\sin\left( \frac {\sqrt{3}}{2}x \right))
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