# ODELS4

Find the particular solution which vanishes at $t=0\,$ and identify the Green's matrix $G_{0}(t,s)\,$ of the system ${\begin{cases}x_{1}'=x_{2}+g_{1}(t)\\x_{2}'=-x_{1}+g_{2}(t)\end{cases}}\,$.

The homogeneous equations can be written ${\vec x}'=A(x){\vec x}\,$ where

$A={\begin{bmatrix}0&1\\-1&0\end{bmatrix}}\,$

Find the eigenvalues $\lambda =\pm i\,$.

The eigenvector for $\lambda _{1}=i\,$ is ${\vec {u_{1}}}={\begin{bmatrix}1\\i\end{bmatrix}}\,$.

The real and imaginary parts are ${\vec v}={\begin{bmatrix}1\\0\end{bmatrix}},{\vec w}={\begin{bmatrix}0\\1\end{bmatrix}}\,$.

So if $\lambda =\alpha +i\beta \,$ then

$e^{{\lambda t}}=e^{{\alpha t}}(\cos \beta t+i\sin \beta t)\,$

${\vec x}=e^{{\lambda t}}{\vec {u_{1}}}\,$

${\vec x}=e^{0}(\cos t+i\sin t){\begin{bmatrix}1\\i\end{bmatrix}}\,$

${\vec x}={\begin{bmatrix}\cos t+i\sin t\\-\sin t+i\cos t\end{bmatrix}}\,$

So two linearly independent solutions are the real and imaginary parts of ${\vec x}\,$.

$x_{1}={\begin{bmatrix}\cos t\\-\sin t\end{bmatrix}},x_{2}={\begin{bmatrix}\sin t\\\cos t\end{bmatrix}}\,$

So the fundamental matrix is

$X(t)={\begin{bmatrix}\cos t&\sin t\\-\sin t&\cos t\end{bmatrix}}\,$

Now the particular solution is

$x_{p}=X(t)\int _{0}^{t}X^{{-1}}(s)g(s)ds\,$

Define Green's matrix

$G_{0}(t,s)={\begin{cases}0&t\leq s\leq b\\X(t)X^{{-1}}(s)&a\leq s

$={\begin{bmatrix}\cos t\cos s+\sin t\sin s&-\cos t\sin s+\sin t\cos s\\-\sin t\cos s+\cos t\sin s&\sin t\sin s+\cos t\cos s\end{bmatrix}}\,$