User:DemonThing/Sandbox

Question

 * 1) $$\vec x'(t) = P(t) \vec x(t) + \vec g(t)$$
 * 2) *$$P(t) = \begin{pmatrix} -2 & 6 \\ 6 & -2 \end{pmatrix}$$
 * 3) *$$\vec g(t) = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$$
 * 4) $$\vec x(0) = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$$

Solution

 * 1) $$\vec x_h(t) = \Psi(t) \vec c$$
 * 2) *$$\Psi(t) = \begin{pmatrix} e^{-8t} & e^{4t} \\ -e^{-8t} & e^{4t} \end{pmatrix}$$
 * 3) $$\vec x(t) = \Psi(t) \vec u(t)$$
 * 4) $$\vec x(t) = \Psi(t) \vec c + \Psi(t) \int{ \Psi^{-1}(t) \vec g(t) \;dt}$$
 * 5) *$$\Psi^{-1}(t) = \frac{1}{2} \begin{pmatrix} e^{8t} & -e^{8t} \\ e^{-4t} & e^{-4t} \end{pmatrix}$$
 * 6) $$\vec x(t) = \Psi(t) \vec c + \Psi(t) \int{ \frac{1}{2} \begin{pmatrix} e^{8t} & -e^{8t} \\ e^{-4t} & e^{-4t} \end{pmatrix} \begin{pmatrix} 1 \\ 1 \end{pmatrix} \;dt}$$
 * 7) $$\vec x(t) = \Psi(t) \vec c + \Psi(t) \int{ \begin{pmatrix} 0 \\ e^{-4t} \end{pmatrix} \;dt}$$
 * 8) $$\vec x(t) = \Psi(t) \vec c + \begin{pmatrix} e^{-8t} & e^{4t} \\ -e^{-8t} & e^{4t} \end{pmatrix} \begin{pmatrix} 0 \\ -\frac{1}{4}e^{-4t}\end{pmatrix}$$
 * 9) $$\vec x(t) = \Psi(t) \vec c + \begin{pmatrix} -\frac{1}{4} \\ -\frac{1}{4} \end{pmatrix}$$
 * 10) $$\vec x(0) = \Psi(0) \vec c + \begin{pmatrix} -\frac{1}{4} \\ -\frac{1}{4} \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$$
 * 11) $$\vec c = \Psi^{-1}(0) \begin{pmatrix} \frac{1}{4} \\ \frac{1}{4} \end{pmatrix} = \frac{1}{2} \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} \frac{1}{4} \\ \frac{1}{4} \end{pmatrix} = \begin{pmatrix} 0 \\ \frac{1}{4} \end{pmatrix}$$
 * 12) $$\vec x(t) = \Psi(t) \vec c + \begin{pmatrix} -\frac{1}{4} \\ -\frac{1}{4} \end{pmatrix} = \begin{pmatrix} e^{-8t} & e^{4t} \\ -e^{-8t} & e^{4t} \end{pmatrix} \begin{pmatrix} 0 \\ \frac{1}{4} \end{pmatrix} + \begin{pmatrix} -\frac{1}{4} \\ -\frac{1}{4} \end{pmatrix} = \begin{pmatrix} \frac{1}{4} e^{4t} - \frac{1}{4} \\ \frac{1}{4} e^{4t} - \frac{1}{4} \end{pmatrix}$$

lol math
Prove $$\sum_{n=-\infty}^\infty \frac{1}{x-n\pi} = \cot x$$