Sketching solutions for 2x2 First order System w/ constant coefficients

June 16, 2019

Let $\begin{align} x'&=-x+by\\ y'&=cx-3y \end{align}$ The $x$ and $y$ are departures from normal advertising budgets for Mass. states and NH. First let’s try $b=2,c=0$. Then we solve for solutions: $\begin{bmatrix} x\\ y \end{bmatrix}=c_1\begin{bmatrix} 1\\ -1 \end{bmatrix}e^{-3t}+c_2\begin{bmatrix} 1\\ 0 \end{bmatrix}e^{-t}$

To plot the solutions, let’s first start with easy solutions. The easy ones are $c_1=\pm 1,\ c_2=0$ and $c_1=0,\ c_2=\pm 1$.