Diff Eq Undetermined Coefficients

[Fernandez]

Homework Statement

Find the solution for y"+2y'+5y=(e^x)sinx

Homework Equations

The Attempt at a Solution

So far I think I've gotten the solution from the characteristic equation, but I'm having trouble with the particular solution.

For the characteristic equation solution:
y"+2y'+5y=0
r^2+2r+5=0
Using the quadratic formula r= -1(+/-)2i
So y=(e^-x)((C1)cos(2x)+(C2)sin(2x)

For the particular solution I think I'm assuming correctly that y=A(e^x)sin(2x)+B(e^x)cos(2x), or am I not catching a term that can be combined with the complementary solution? Or am I just using the wrong equation all together?

 

[G01]

In this line: y=A(e^x)sin(2x)+B(e^x)cos(2x), you shouldn't have the 2's inside the trig arguments. Since the RHS of the problem statement has only a sin(x), your "guess" should only include trig functions with x as the argument. You are getting the 2's from the Homogeneous Solution, but in this problem, the H solution, does not affect your "guess" of the particular solution.

 

[Fernandez]

Thanks! I don't know how I overlooked that.