Differential Equation Suppose Ms. Lee is buying a new house and must borrow 150,000. She wants a 
30-year mortgage and she has two choices. She can either borrow money at 7% per 
year with no points, or she can borrow the money at 6.5% per year with a charge of 
3 points. (A "point" is a fee of 1% of the loan amount that the borrower pays the 
lender at the beginning of the loan. For example, a mortgage with 3 points requires 
Ms. Lee to pay 4,500 extra to get the loan.) As an approximation, we assume that 
interest is compounded and payments are made continuously. Let 
$$M(t) = \text{amount owed at time } t\ \left(\text{measured in years}\right)$$
$$r= \text{annual interest rate, and}$$
$$p= \text{annual payment}$$
Then the model for the amount owed is   
$$ \frac{dM}{dt}=rM-p$$
Q.How much does Ms Lee has to pay in each case?  
I have tried solving the DE, and i get
$$ M(t)=C_1e^{rt} + \frac{p}{r}$$  
Now what to do?
 A: We can work with concrete numbers, or develop a general formula.  Ideally, you should do both, as an exercise and a partial check.  Let's develop a general formula. I will use your notation, but introduce two new symbols. Let $N$ be the amortization period, that is, the number of years until the mortgage is paid off. In our case, $N=30$.  Let $A$ be the initial amount owed.  With no "points", $A=150000$.  With $3$ points, she needs to borrow $150000/(1.03)$ in order to have $150000$ left after paying the points. The general equation for the amount owed is, as you wrote,
$$M(t)=C_1e^{rt}+\frac{p}{r}$$
This is the general solution of the equation, but it is incomplete until we evaluate the constant $C_1$. (Technical note: It will turn out, of course, that $C_1$ is negative, else what we owe would increase rapidly forever. I would have preferred to arrange things so that any constant is positive.)
Note that $M(0)=A$ and $M(N)=0$. We obtain the two equations 
$$A=C_1+\frac{p}{r}$$
$$0=C_1e^{rN} +\frac{p}{r}$$
Subtract, to get rid of the $p/r$ term. We get
$$A=C_1(1-e^{rN})$$
So $C_1=-\frac{A}{e^{rN}-1}$ and we obtain the equation
$$M(t)=\frac{p}{r} -\frac{A}{e^{rN}-1}e^{rt}$$
Now that we have full information about $M(t)$, we should be able to answer any question.  In particular, by taking $t=N$, we have
$$0=\frac{p}{r} -\frac{A}{e^{rN}-1}e^{rN}$$
Now we can solve for the payment $p$:
$$p=rA\frac{e^{rN}}{e^{rN}-1}=\frac{rA}{1-e^{-rN}}$$
and easily find $p$ given any  $r$, $N$, and $A$. 
A: To calculate how much she borrows with $3$ points, she gets $0.97$ of the amount borrowed, so $M(0)=150,000/0.97$, which is slightly higher than $154,500$.  Now if you incorporate $M(0)$ and $M(30)$ you can solve for $C_1$ and $p$ and pick the choice with lower $p$
