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Find a fundamental matrix for the following system $y'(t)=Ay(t)$ having the coefficient matrix $A=\begin{bmatrix} 1 & 2 \\ 4 & 3\end{bmatrix}$. Also find the particular solution satisfying the initial condition $\eta=\begin{bmatrix} 3 \\ 3\end{bmatrix}$

Since the fundamental matrix have coloumns $e^{\lambda_it}v_i$, where $\lambda_i$ is the eigenvalues for A and $v_i$ is the eigenvectors of A $(i=1,2)$, i've found the fundamental matrix to be $\Phi(t)=\begin{bmatrix} \frac{1}{2}e^{5t} & -e^{-t} \\ e^{5t} & e^{-t} \end{bmatrix}$

How do I find the particular solution satisfying the initial condition $\eta$?

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    I'd say so, since that's been the case throughout my textbook, but it doesn't mention it in this assignment – 2017-02-15
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    Thanks! i'll take a look – 2017-02-15
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    [This](http://math.stackexchange.com/questions/744296/differential-equations-diagonalizable-matrix/744324#744324) elementary method will find you the solution. – 2017-02-15

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