Answer to Question #272664 in Differential Equations for Lando

Question #272664

(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


1
Expert's answer
2021-11-30T16:46:57-0500

(i)

Existence theorem of the 1st order differential equation;

The general first order ODE isdydx=F(x,y)\frac{dy}{dx}=F(x,y) , y(x0)=y0y(x_0)=y_0 .......(i)

Suppose that F (x,y) is a continuous function defined on some region; R={x,y:x0δ<x<x0+δ,y0ϵ<y<y0+ϵ}R=\{x,y:x_0-\delta<x<x_0+\delta,y_0-\epsilon<y<y_0+\epsilon\}

containing the point (x0,y0)(x_0,y_0) . Then there exists a number δ1\delta_1 (possibly smaller than δ\delta) so that a solution y=f(x)y=f(x) to equation (i) is defined for x0δ1<x<x0+δ1x_0-\delta_1<x<x_0+\delta_1

Uniqueness theorem of the first order differential equation;

Suppose that both F(x,y) and dFdy(x,y)\frac{dF}{dy}(x,y) are continuous functions defined on a region R, R={x,y:x0δ<x<x0+δ,y0ϵ<y<y0+ϵ}R=\{x,y:x_0-\delta<x<x_0+\delta,y_0-\epsilon<y<y_0+\epsilon\}

then there exists a number δ2\delta_2 (possibly smaller than δ1)\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 =0.0712\frac{0.07}{12}

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

M=Pr(1+r)n(1+r)n1M=\frac{P\cdot r(1+r)^n}{(1+r)^n-1}

700=P(0.0712)(1+0.0712)360(1+0.0712)3601700=\frac{P\cdot (\frac{0.07}{12})(1+\frac{0.07}{12})^{360}}{(1+\frac{0.07}{12})^{360}-1}

700=P(0.0712)(8.1164975)8.11649751700=\frac{P\cdot (\frac{0.07}{12})(8.1164975)}{8.1164975-1}

7007.1164975=P(0.0712)(8.1164975)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


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