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
First we need to solve the Homogenous system
To get the complimentary solution
Now we will use the variation of parameters suppose the particular solution is of the form
Now if we plug this into the ODE we get
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
Where the integration is done entrywise.
So finally the particular solution is