Ok first take the characteristic equition x^2-x+1=0 and solve that first.
Another approach would be the annihilator method.
We are given:
(1)
Observe that annihilates and annihilates and so define:
Applying to 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: