$(1$ point$)$ Liam opens a bank account with an initial balance of $500$ dollars. Let $b(t)$ be the balance in the account at time $t.$ Thus $b(0)=500.$ The bank is paying interest at a continuous rate of $5\%$ per year. Liam makes deposits into the account at a continuous rate of $s(t)$ dollars per year. Suppose that $s(0)=500$ and that $s(t)$ is increasing at a continuous rate of $2\%$ per year $($Liam can save more as his income goes up over time$).$

$($a$)$ Set up a linear system of the form

$\frac{db}{dt}=m_{11}b+m_{12}s$

$\frac{ds}{dt}=m_{21}b+m_{22}s$

$m_{11}=$

$m_{12}=$

$m_{21}=$

$m_{22}=$

$($b$)$ Find $b(t)$ and $s(t)$

$b(t)=$

$s(t)=$