Abstract algebra: systems of differential linear equations

[vikkivi]

Homework Statement

Solve the inhomogeneous differential equation dX/dt=AX+B in terms of the solutions to the homogeneous equation dX/dt=AX.

Homework Equations

A is an nxn real or complex matrix and X(t) is an n-dimensional vector-valued function.
If v is an eigenvector for A with eigenvalue a, then X=v*e^a*t is a particular solution to the differential equation dX/dt=AX.
And the general solution of the homogenous eqn is X=P^-1*Xtilda.

The Attempt at a Solution

So, the general solution of the inhomogeneous equation should be a particular solution of the inhomogenous equations + the general solution of the homogeneous equation. We know the general part, but I am lost on how to find the particular solution for an inhomogeneous equation. Any help would be apprecaited!

 

[CompuChip]

Is B a constant vector? In that case, you could try a solution where X is also a constant vector, such that dX/dt = 0.

 

[HallsofIvy]

You can use a multi-dimensional version of "variation of parameters". If the general solution to the homogeneous equation is X= P^{-1}X^~ try a solution of the from Y= P^{-1}X^~u(t) where u(t) is an unknown function. Then Y'= P^{-1}X^~'u+ P^{-1}X^~u' and Ay= AP^{-1}X^~u so the equation becomes P^{-1}X^~'u+ P^{-1}X^~u'+ AP^{-1}X^~u= B.

Since X= P^{-1}X^~ is a solution to the homogeneous equation, P^{-1}X^~'u+ AP^{-1}X^~u= (P^{-1}X^~'+ AP^{-1}X^~)u= 0 and the equation reduces to P^{-1}X^~u'= B so u'= X^~^{-1}PB. Integrate that to find u.