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#### karush

##### Well-known member

- Jan 31, 2012

- 2,886

Solve the initial value problem

$Y'=\left|\begin{array}{rr}2 & 1 \\-1 & 2 \end{array}\right|Y

+\left|\begin{array}{rr}e^x \\0 \end{array}\right|,

\quad Y(0)=\left|\begin{array}{rr} 1 \\1 \end{array}\right| $

ok so we have the form $y'=AY+G$

rewrite as

$$\displaystyle

\left|\begin{array}{rr}y_1^\prime \\y_2^\prime \end{array}\right|

=\left|\begin{array}{rr}2 & 1 \\-1 & 2 \end{array}\right|

\left|\begin{array}{rr}y_1 \\y_2\end{array}\right|

+\left|\begin{array}{rr}e^x \\0 \end{array}\right|$$

ok so the next thing to do is find eigenvalues of A so

$\left| \begin{array}{cc}

-\lambda+2&1\\-1&-\lambda+2\end{array}

\right|

=\left(-\lambda+2\right)^{2}+1$

so roots are

$\lambda_{1}=2 + i, \qquad \lambda_{2}=2 - i$

so far ???? hopefully