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I've come across a problem in my differential equations book that I can't seem to be able to solve (it's not a homework problem, I'm just practicing):

"Using matrix algebra techniques and the method of undetermined coefficients, find a general solution for

x''(t) + y'(t) - x(t) + y(t) = -1,

x'(t) + y'(t) - x(t) = t^2 "

At first, I tried to change this system to a system of first-order differential functions only using the following substitutions:

x1(t) = x(t)

x2(t) = x'(t)

x3(t) = y(t)

Which leads to the following system in matrix form:

[itex]\left[ \begin{array}{c} x1' \\ x2' \\ x3' \end{array} \right] = \left[ \begin{array}{ccc} 0 & 1 & 0 \\ 0 & 1 & -1 \\ 1 & -1 & 0 \end{array} \right] * \left[ \begin{array}{c} x1 \\ x2 \\ x3 \end{array} \right] + \left[ \begin{array}{c} 0 \\ -t^2 - 1 \\ t^2 \end{array} \right] [/itex]

Next, I solved the system of heterogeneous equations associated with this system, which also wasn't a problem. But now I'm stuck:

The nonhomogeneous part suggests a particular solution of the form [itex]Bt^2+Ct+D[/itex], with B, C, and D being vectors to be determined. Putting everything into the system yields

[itex]2*Bt+C = A*(Bt^2 + Ct + D) + \left[ \begin{array}{c} 0 \\ -t^2-1 \\ t^2 \end{array} \right][/itex], with A being the coefficient matrix of my original system. But how do I proceed from here?

Thanks,

Alex

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# Solving system of differential equations using undetermined coefficients

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