Question $3$

The mass of chemical in each tank is given by the following system of ODEs:

$y_1=(5+R)(-\frac{y_1}{10}+\frac{y_2}{50})$

$y_2=(5+R)(\frac{y_1}{10}-\frac{y_2}{50})$

Where $y_1$ and $y_2$ are the masses $($in grams$)$ of chemical in tank $1$ and tank $2$ respectively as functions of time. $R$ is the $6^{th}$ digit of your URN. For example, if your URN is $6835725$ then $R=2.$

a$)$ Using your value of $R,$ write this system of ODEs in matrix form.

$2$

b$)$ Show that the following is a solution to this system of ODEs $($using your value of R$)$:

$\left[\begin{array}{c}{y}_1\\ y_2\end{array}\right]=(3+R)\left[\begin{array}{c}1\\ 5\end{array}\right]+3e^{-(\frac{3}{25})(5+R)t}\left[\begin{array}{c}-1\\ 1\end{array}\right]$

c$)$ What initial conditions must be used to obtain the solution given in part $($b$)?$