- #1

Eclair_de_XII

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## Homework Statement

"Someone with no initial capital ##S_0=0## invests ##k \frac{dollar}{year}## at an annual rate of return ##r##. Assume that investments are made continuously and that the return is compounded continuously."

Find the solution ##S## that solves the differential equation modeling this scenario.

## Homework Equations

Let ##S## be the amount of capital at any time.

## The Attempt at a Solution

I'm trying to make sure that the units of all my quantities are the same, but the main problem I'm having is the discontinuity at ##t=0## when I divide ##S## by ##t##.

##\frac{dS}{dt}=\frac{S}{t}+k(r+1)##

##\frac{dS}{dt}-\frac{S}{t}=k(r+1)##

##\frac{d}{dt}(t^{-1}S)=\frac{k(r+1)}{t}##

##S(t)=(tlnt)(k(r+1))##

I frankly do not know how to model this problem. Can anyone help me understand what is happening in this problem? It's hard to insert ##S## into the equation because the ##\frac{dS}{dt}## does not rely on ##S##; it relies on the rate of return ##r## which only applies to the yearly investments ##k##. Then it's compounded continually...? I do not understand this.