In this problem you will solve the nonhomogeneous system $$\boldsymbol{\vec{y}\,'}\ = \left\lbrack \begin{array}{rr} 4 & -2 \\ 5 & -2 \end{array} \right\rbrack \, \boldsymbol{\vec{y}} \ + \ \left\lbrack \begin{array}{r} -3e^{t} \\ 2e^{t} \end{array} \right\rbrack$$

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A. Write a fundamental matrix for the associated homogeneous system

$\boldsymbol{\Psi}\ =\ $

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

B. Compute the inverse

$\boldsymbol{\Psi}^{\mathrm{-}1}\ =\ $

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

C. Multiply by $\boldsymbol{\vec{g}}$ and integrate

$\displaystyle \int \boldsymbol{\Psi}^{\mathrm{-}1}\; \boldsymbol{\vec{g}}\; dt\ =\ $

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $+ c_1$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$ $+ c_2$

(Do not include $c_1$ and $c_2$ in your answers).

D. Give the solution to the system

$\displaystyle \boldsymbol{\vec{y}}\ =\ $

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

$c_1 +$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

$c_2$

$\phantom{y = } \quad +$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

(Do not include $c_1$ and $c_2$ in your answers).

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If you don't get this in 2 tries, you can get a hint.

Hint: