This is a direct generalization of solving inhomogeneous ODE'sWhen I got 8 (first order) differential equations. I need particular solutions, and in past problem with wronskian, constants had differentiation. I should integrate all of them. When I integrate 4th and 8th equation, I have f1[t] and f2[t]. I know that solution of diff equation of 4th order had form x=C1*e^(i*p1*t)+C2*e^(-i*p1*t)+C3*e^(i*p2*t)+C4*e^(-i*p2*t)+C5*e^(p3*t)+C6*e^(-p3*t)+C7*e^(p4*t)+C8*e^(-p4*t) or x=C1*sin(p1*t)+C2*cos(p2*t)+C3*sinh(p3*t)+C4*cosh( p4*t). How to find particular solutions of system? Revise me about premise above...
To solve the system
$\displaystyle \mathbf{X}'=A\mathbf{X}+\mathbf{F}(t)$
First we need to solve the Homogenous system
$\displaystyle \mathbf{X}'=A\mathbf{X}$
To get the complimentary solution $\displaystyle \mathbf{X}_c$
Now we will use the variation of parameters suppose the particular solution is of the form
$\displaystyle \displaystyle \mathbf{X}_p=\mathbf{X}_c\mathbf{U}(t), \text{ where } \mathbf{U}(t)=\begin{pmatrix} u_1(t) \\ u_2(t) \\ u_3(t) \\ u_4(t) \\ u_5(t) \\ u_6(t) \\ u_7(t) \\ u_8(t)\end{pmatrix}$
Now if we plug this into the ODE we get
$\displaystyle \mathbf{X}'_c\mathbf{U}(t)+\mathbf{X}_c\mathbf{U}' (t)=A\mathbf{X}_c\mathbf{U}(t)+\mathbf{F}(t)$
This gives
$\displaystyle [\mathbf{X}'_c-A\mathbf{X}_c]\mathbf{U}(t)+\mathbf{X}_c\mathbf{U}'(t)=\mathbf{F }(t) \iff \mathbf{X}_c\mathbf{U}'(t)=\mathbf{F}(t) $
The first group is zero because it is a solution to the homogenous system
Since the complimentary solution is a fundamental matrix it's determinant is not always zero so it is invertable this gives
$\displaystyle \displaystyle \mathbf{X}_c\mathbf{U}'(t)=\mathbf{F}(t) \iff \mathbf{U}'(t)=\mathbf{X}^{-1}_c\mathbf{F}(t) \iff \mathbf{U}(t)=\int \mathbf{X}^{-1}_c\mathbf{F}(t)dt$
Where the integration is done entrywise.
So finally the particular solution is
$\displaystyle \displaystyle \mathbf{X}_p=\mathbf{X}_c\int \mathbf{X}^{-1}_c\mathbf{F}(t)dt$