Help with solving system of DE's

[gkirkland]

My DE skills are a bit rusty, and I need some help remembering how to handle a system such as:

$\dot{x_1}=x_2$
$\dot{x_2}=-2x_1-3x_2+sint+e^t$

I have found the homogeneous solution to be (sorry I don't know how to do matrices here):

$c_1\left\{e^{-t}\right\}+c_2\left\{e^{-2t}\right\}$
$c_1\left\{-e^{-t}\right\}+c_2\left\{-2e^{-2t}\right\}$

From what I've found online I should guess a particular solution form:

$x_{p}=Asin(t)+Bcos(t)+Ce^{t}$

Where A, B, and C are 2x1 matrices of constants $a_1, a_2, b_1, b_2, c_1, c_2$

Is this correct?
Then rewrite the original in the form:

$\dot{x_{p}}=Ax_{p}+g$

Then differentiate the guess and substitute back into the above.

Assuming this is all correct, what are the next steps in finding the general solution?

 

[matematikawan]

Assuming that your system can be written as the matrix form $\dot{X}=AX+F(t).$ , e.g. X=[x₁ x₂]^t etc
Then the general solution for this equation should be (if I'm not mistaken)
$$X(t)=e^{At}C+e^{At}\int_0^t e^{-As}F(s) ds$$

Your particular solution in this case is then
$$X_p(t)=e^{At}\int_0^t e^{-As}F(s) ds$$