I'm supposed to derive a formula for mortgages, $f(t)$, which represents the current debt at time $t$. It's given that ($m$ and $r$ are constants):
$\displaystyle \frac{df(t)}{dt}= r\cdot f(t) - 12m$, I've found the solutions $\displaystyle f(t) = K\cdot e^{rt} +\frac{12m}{r}$ (With $K, m$ and $r$ as constants).
But all of these solutions clash with the fact that $f(N)=0$ and $f(0)=B, \ B \ne 0$.
Solving for $K$ in the term $\displaystyle f(N)=0 =K\cdot e^{rt}+\frac{12m}{r}$ yields zero for every possible input to the function $f$, while it's only supposed to yield $0$ for $t=N$ and it's supposed to be at a maximum for $t=0$, what am I doing wrong here?
By solving the ODE is simple to find that $\displaystyle f(t)=\frac{1}{r}e^{rt}+\frac{12m}{r}$.
We want $f(0)=B \rightarrow \displaystyle \frac{1}{r}e^{0}+\frac{12m}{r}=B \Rightarrow \displaystyle m=\frac{Br-1}{12}$.
We got $\displaystyle f(t)=\frac{1}{r}e^{rt}+\frac{Br-1}{r}$.
Now we want $\displaystyle f(N)=0$, so $$\displaystyle \frac{1}{r}e^{rN}+\frac{Br-1}{r}=0 \rightarrow e^{rN}+Br-1=0 \rightarrow rN=\ln(1-Br) \Rightarrow N=\frac{\ln(1-Br)}{r}$$
You asked for B in terms of N, so we can write $\displaystyle Br=1-e^{rN} \Rightarrow B=\frac{1-e^{rN}}{r}$.
Hope this answer helps.
If the problem is solving the ODE, I would love to help.