1.

By solving the initial value problem

y+3y=12+4sin2t, y(0)=0.

We got the solution is y=4+4/13(3sin2t-2cos2t)-44/13e^-3t

Describe the behavior of the solution for large values of t ?

2.

I borrow P₀ dollars from a lender at a per annum interest rate of r compounded continuously. I pay my loan off continuously at a constant rate of k dollars per annum.

(a) If P(t) is the money I owe in dollars as a function of time t (measured in years), set up the initial value problem for which P (t) is the solution. Then solve this problem, express- ing P(t) in terms of P₀,t,r and k.

(b) What should k be if I need to pay my loan off in T years ? Express your answer in terms of P₀,r and T.

Plese type instead of hand-writing. If you cannot type, please write legibal otherwise I cannot read it. Thank you