Question

Solving a system of linear differential equations; don't need help yet, posting below.

Answer 1

\[\frac{dx}{dt}=x-2y\]\[\frac{dy}{dt}=5x-y\]\[x(0)=-1, \ \ \ y(0)=0.\]

Answer 2

\[\mathcal{L}\left\{x'\right\}=\mathcal{L}\left\{x \right\}-2\mathcal{L}\left\{y\right\}\]\[sX(s)-x(0)=X(s)-2Y(s);\]
\[\mathcal{L}\left\{y'\right\} = 5 \mathcal{L}\left\{x\right\}-\mathcal{L}\left\{y\right\}\]\[sY(s)-y(0)=5X(s)-Y(s).\]

Answer 3

\[X(s)(s-1)=2Y(s)-1;\]\[Y(s)(s+1)=5X(s)+2.\]
//ERRATA: y(0)=2.

Answer 4

\[X(s)=\frac{2Y(s)-1}{s-1}, \ \ \ Y(s)=\frac{5X(s)+2}{s+1}\]

Answer 5

\[X(s)=\frac{2[5X(s)+2)/(s+1)]-1}{(s-1)}\]

Answer 6

\[X(s)=\frac{10X(s)+4}{(s-1)(s+1)}-\frac{(s+1)}{(s-1)(s+1)}\]

Answer 7

\[X(s)\bigg(\frac{(s-1)(s+1)-10}{(s-1)(s+1)}\Bigg)=\frac{4-(s+1)}{(s-1)(s+1)}\]

Answer 8

\[X(s)=\bigg[\frac{4-(s+1)}{(s-1)(s+1)}\bigg].\bigg[\frac{(s-1)(s+1)}{(s-1)(s+1)-10}\bigg]\]
Cancelling the numerator and denominator of the right and left respectively,

Answer 9

\[X(s)=\frac{4-(s+1)}{(s-1)(s+1)-10}=\frac{3-s}{s^2-11}\]

Answer 10

\[X(s)=\frac{3}{s^2-11}-\frac{s}{s^2-11}\]

Answer 11

\[x(t)=\mathcal{L^{-1}}\left\{\vphantom{}\frac{3}{s^2-11}\right\}-\mathcal{L^{-1}}\left\{\vphantom{}\frac{s}{s^2-11}\right\}\]

Answer 12

\[x(t)=\frac{3}{\sqrt{11}}\mathcal{L^{-1}}\left\{\vphantom{}\frac{\sqrt{11}}{s^2-11}\right\}-\mathcal{L^{-1}}\left\{\vphantom{}\frac{s}{s^2-11}\right\}\]

Answer 13

\[x(t)=\frac{3}{\sqrt{11}}\sinh(\sqrt{11}t)-\cosh(\sqrt{11}t)\]

Answer 14

http://www.wolframalpha.com/input/?i=x%27+%3D+x-2y%2C+y%27+%3D+5x-y%2C+x%280%29%3D-1%2Cy%280%29%3D2