Hey,

I'm stuck on this one and it's due tomorrow so I'd love some help.

Solve y''-y'+y=x+e^x

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- January 8th 2013, 07:24 AMYahikkoA second grade differential equation
Hey,

I'm stuck on this one and it's due tomorrow so I'd love some help.

Solve y''-y'+y=x+e^x - January 8th 2013, 07:40 AMSMADRe: A second grade differential equation
Ok first take the characteristic equition x^2-x+1=0 and solve that first.

- January 8th 2013, 12:37 PMMarkFLRe: A second grade differential equation
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: