# Thread: Word problem with DE

1. ## Word problem with DE

Ok, I simply cannot get the answer that they have in the solutions manual, nor do they explain how they got the answer, so I really have no clue what I'm doing wrong.

The instructions for the problem are: Investment. A brokerage firm opens a new real estate investment plan for which the earnings are equivalent to continuous compounding at the rate of $r$. The firm estimates that deposits from investors will create a net cash flow of $Pt$ dollars, where $t$ is the time in years. The rate of change in the total investment $A$ is modeled by

$\frac{dA}{dt}=rA+Pt$

(a) Solve the differential equation and find the total investment $A$ as a function of $t$. Assume that $A = 0$ when $t = 0$.

(b) Find the total investment A after 10 years given that P = $500,000 and r = 9%. Ok, so I rearranged the DE to be $A' - rA = Pt$ and then determined that $P(t) = -r$ and $Q(t)=Pt$. From there, I found $IF=e^{\int-rdt}=e^{-rt}$. I then mutiplied each side by the IF to get $e^{-rt}A'-rAe^{-rt}=Pte^{-rt}$. From there I doublechecked that the IF was correct, it was, and then I did $\int \frac{dA}{dt} (e^{-rt} * A) = \int Pte^{-rt}$ The integral on the left cancels with the derivative so $e^{-rt}A=\int Pte^{-rt}$ I then pulled P out as a constant to get $e^{-rt}A=P\int te^{-rt}$ I then did parts, using $u=t$, $du=dt$, $V=e^{-rt}$ to get $e^{-rt}A=P(te^{-rt} - \int e^{-rt}dt)$ From there I used $u=-rt$ and $du=-rdt$ and since I only have a $dt$ I did $-\frac{1}{r}du=dt$ Solving the final integral I came up with $e^{-rt}A=P(te^{-rt}+\frac{1}{r}e^{-rt}+C)$ Dividing both sides by $e^{-rt}$ to get $A$ by itself, I got $A=P(t+\frac{1}{r}+Ce^{rt})$ Which is different from what the solutions manual got, which was $A=\frac{P}{r^2}(-rt-1+Ce^{rt})$ What am I doing wrong? 2. Originally Posted by emttim84 Ok, I simply cannot get the answer that they have in the solutions manual, nor do they explain how they got the answer, so I really have no clue what I'm doing wrong. The instructions for the problem are: Investment. A brokerage firm opens a new real estate investment plan for which the earnings are equivalent to continuous compounding at the rate of $r$. The firm estimates that deposits from investors will create a net cash flow of $Pt$ dollars, where $t$ is the time in years. The rate of change in the total investment $A$ is modeled by $\frac{dA}{dt}=rA+Pt$ (a) Solve the differential equation and find the total investment $A$ as a function of $t$. Assume that $A = 0$ when $t = 0$. (b) Find the total investment A after 10 years given that P =$500,000 and r = 9%.

Ok, so I rearranged the DE to be $A' - rA = Pt$ and then determined that $P(t) = -r$ and $Q(t)=Pt$. From there, I found $IF=e^{\int-rdt}=e^{-rt}$.

I then mutiplied each side by the IF to get $e^{-rt}A'-rAe^{-rt}=Pte^{-rt}$.

From there I doublechecked that the IF was correct, it was, and then I did $\int \frac{dA}{dt} (e^{-rt} * A) = \int Pte^{-rt}$

The integral on the left cancels with the derivative so

$e^{-rt}A=\int Pte^{-rt}$

I then pulled P out as a constant to get $e^{-rt}A=P\int te^{-rt}$

I then did parts, using $u=t$, $du=dt$, $V=e^{-rt}$ to get

$e^{-rt}A=P(te^{-rt} - \int e^{-rt}dt)$

From there I used $u=-rt$ and $du=-rdt$ and since I only have a $dt$ I did $-\frac{1}{r}du=dt$

Solving the final integral I came up with

$e^{-rt}A=P(te^{-rt}+\frac{1}{r}e^{-rt}+C)$

Dividing both sides by $e^{-rt}$ to get $A$ by itself, I got

$A=P(t+\frac{1}{r}+Ce^{rt})$

Which is different from what the solutions manual got, which was

$A=\frac{P}{r^2}(-rt-1+Ce^{rt})$

What am I doing wrong?
Is P a function or a variable or a constant? if it is a constant this is a simple seperable differential equation

3. your integration by parts is incorrect
the correct integration is

$\int Pte^{-rt}=P\left[ -\frac{t}{r}e^{-rt}-\frac{1}{r^2}e^{-rt}+C \right]$

4. Originally Posted by TheEmptySet
your integration by parts is incorrect
the correct integration is

$\int Pte^{-rt}=P\left[ -\frac{t}{r}e^{-rt}-\frac{1}{r^2}e^{-rt}+C \right]$
Right, I figured the integration is incorrect, but where is the error so I can see how to get the correct answer, and not just have the correct answer?