- #1

Drakkith

Staff Emeritus

Science Advisor

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## Homework Statement

A Solve the following initial value problem:

##\frac{dx}{dt}=-x(1-x)##

##x(0)=\frac{3}{2}##

B. At what finite time does ##x→∞##

## Homework Equations

## The Attempt at a Solution

##\frac{dx}{dt}=x(x-1)##

##\frac{dx}{x(x-1)}=dt##

Partial fractions: ##dx(\frac{-1}{x}-\frac{1}{x-1})=dt##

Integrating both sides: ##ln|\frac{1}{x}|-ln|x-1|=t+c##

##ln|\frac{1}{x(x-1)}|=t+c##

e to the power of both sides and taking the constant ##e^c## as A: ##\frac{1}{x(x-1)}=Ae^t##

Plugging in the initial value gives me ##A=\frac{4}{3}##

My final equation is: ##\frac{1}{x(x-1)} = \frac{4}{3}e^t##

What I don't understand is how ##x## is related to ##t## and how to figure out at what time x goes to infinity. Offhand I don't see any way for X to increase to infinity since t is an exponent of e, unless t goes to negative infinity.