Question

Can anyone please help me out with this question??

Answer 1

Here is the question :
For the system if inhomogeneous differential equation:
\[\frac{ dx }{ dt }=5x-y+2 and
\frac{ dy }{ dt }= x+y-4t\]
and the initaial condition X(0)=1 and Y(0)=2:
A) write the system in the form of
\[\frac{ du(t) }{ dt}=[A]u(t)+b(t)\]
b)find the jordan form [J] and matrix [M] such that [J]=[M][J][M]^-1
c)find the matrix exponential \[e^{At}\]
d)use the initial condition to find the solution x(t) and y(t)

Answer 2

Let \[u(t)=\left(\begin{matrix}x \\y \end{matrix}\right)\] so that \[u'(t)=\left(\begin{matrix}x' \\ y'\end{matrix}\right)\] Then it follows by the definitions of matrix products and the given system that \[u'(t)=[A]u(t)+b(t)\] where \[A=\left[\begin{matrix}5 & -1 \\ 1 & 1\end{matrix}\right], \quad b(t)=\left(\begin{matrix}-4t\\0\end{matrix}\right)\] That's step 1.

I assume that you're supposed to find the Jordan canonical form of A, correct?