# Science:Math Exam Resources/Courses/MATH105/April 2010/Question 05 (b)/Solution 2

From part (a) we have that

${\displaystyle {\frac {{\textrm {d}}y}{{\textrm {d}}t}}=0.05y(t)-A}$

with the initial amount of money owed being ${\displaystyle y(0)=240000}$. The goal is to solve this equation for the rate of ${\displaystyle y}$ so that we can actually obtain the quantity we want, ${\displaystyle y(t)}$ which is the money owed after ${\displaystyle t}$ years. Consider the substitution,

${\displaystyle \displaystyle {}u(t)=0.05y-A}$

which implies that

${\displaystyle {\frac {{\textrm {d}}u}{{\textrm {d}}t}}=0.05{\frac {{\textrm {d}}y}{{\textrm {d}}t}}.}$

Therefore we can rewrite (and simplify) our differential equation as

${\displaystyle {\frac {{\textrm {d}}u}{{\textrm {d}}t}}=ru}$

which we recognize as the exponential model and so we know the solution is

${\displaystyle \displaystyle {}u(t)=B\exp(0.05t)}$

with ${\displaystyle B}$ an arbitrary constant. This solution can also be determined from separation of variables and integration. Putting back into our original variable, ${\displaystyle y(t)}$, we get

${\displaystyle y(t)={\frac {B\exp(0.05t)+A}{0.05}}.}$

which is our desired equation for the amount owed on the mortgage after time ${\displaystyle t}$. Using the initial condition

${\displaystyle y(0)={\frac {B+A}{0.05}}=240000}$

we get that ${\displaystyle B=12000-A}$. Therefore we are able to write the amount of money owed solely in terms of the annual payments, ${\displaystyle A}$,

${\displaystyle y(t)={\frac {(12000-A)\exp(0.05t)+A}{0.05}}.}$

Now when ${\displaystyle t=25}$, we want that the mortgage is paid off, i.e., we want the amount of money we owe to be zero. Therefore we seek that ${\displaystyle y(25)=0}$. Therefore,

{\displaystyle {\begin{aligned}0&={\frac {(12000-A)\exp(0.05\cdot 25)}{0.05}}\\A&={\frac {12000\exp(0.05\cdot 25)}{\exp(0.05\cdot 25)-1}}=16818.61.\end{aligned}}}

Therefore, in order to pay off the mortgage in 25 years, we require that ${\displaystyle A}$ = 16818.61. Recall that when we started paying the mortgage at year zero we owed 80% of the value or 240 000 dollars. If we want to find out how much we really paid, we just need to multiply our annual rate, ${\displaystyle A}$ by 25 years to get $420 465.25. Add this to the$60 000 we already paid up front then our $300 000 dollar house has actually cost us$480 465.25!