Consider the differential equation

$y^{\text{'}}-\frac{3}{2}y=3t2e^t$

$($a$)$ Draw a slope field for tin$[-1,5]$ and yin$[-3,3].(2$ point$)$

$($b$)$ Find the general solution. Be sure to write out all steps. $($Hint: you can use integrating factor and integration by parts$)(2$ points$)$

$($c$)$ Describe how solutions behave as $t.$ How do you know? $(1$ point$)$

$2.$ Sally currently has $$60,000$ worth of student loans. The loans have a fixed interest rate of $5\%$ compounded continuously and Sally pays $$4,000$ every year.

$($a$)$ Write the differential equation for remaining principal $($i$.$e$.$ how much Sally still owes$),P,$ after $t$ years. $(1$ point$)$

$($b$)$ Find the solution for the remaining principal after $t$ years. Assume $P(0)=600000.(1$ point$)$

$($c$)$ How long will it take for Sally to pay off her student loans, i$.$e$.P=0?$ Show your work. $(1$ points$)$

$($d$)$ Suppose Sally paid $$1,000$ every year instead. When will she pay off her student loans? Show your work and justify your answer. $(2$ points$)$

$($e$)$ Extra credit: What is the critical value that Sally must pay every year so that paying off her student loans is even possible? Show your work. $(1$ points$)$