Answer to Question #272664 in Differential Equations for Lando

Question

(i) State the Existence and Uniqueness theorem for the

differential equation of the first order.

(ii) A home buyer can spend no more than $700 per month on

mortgage payments. Suppose that the interest rate is

7% and that the term of the mortgage is 30 years.

Assume that the interest is compounded continuously

and that payments are also made continuously.

i. Determine the maximum amount that this buyer can

borrow.

ii. Determine the total interest paid during the term

of the mortgage

Answer 1

(i)

Existence theorem of the 1st order differential equation;

The general first order ODE is$\frac{dy}{dx}=F(x,y)$ , $y(x_0)=y_0$ .......(i)

Suppose that F (x,y) is a continuous function defined on some region; $R=\{x,y:x_0-\delta<x<x_0+\delta,y_0-\epsilon<y<y_0+\epsilon\}$

containing the point $(x_0,y_0)$ . Then there exists a number $\delta_1$ (possibly smaller than $\delta$) so that a solution $y=f(x)$ to equation (i) is defined for $x_0-\delta_1<x<x_0+\delta_1$

Uniqueness theorem of the first order differential equation;

Suppose that both F(x,y) and $\frac{dF}{dy}(x,y)$ are continuous functions defined on a region R, $R=\{x,y:x_0-\delta<x<x_0+\delta,y_0-\epsilon<y<y_0+\epsilon\}$

then there exists a number $\delta_2$ (possibly smaller than $\delta_1)$ so that the solution y=f(x) to equation (i) is the unique solution.

(ii)

(i) m=700

r=7%=0.07

Rate per month =$\frac{0.07}{12}$

n=30yrs=(30*12)months=360months

$M=\frac{P\cdot r(1+r)^n}{(1+r)^n-1}$

$700=\frac{P\cdot (\frac{0.07}{12})(1+\frac{0.07}{12})^{360}}{(1+\frac{0.07}{12})^{360}-1}$

$700=\frac{P\cdot (\frac{0.07}{12})(8.1164975)}{8.1164975-1}$

$700*7.1164975=P(\frac{0.07}{12})(8.1164975)$

0.047346P=4981.5482

P=$105215.81

(ii)I=T-P

T=m*n

=700*360

=252000.00

$\therefore$ I=252000.00-105215.81

=$146784.19