# Thread: Differential Equations and exponential growth

1. ## Differential Equations and exponential growth

The volume of water in a solution during a chemical process varies with time and satisfies the equation: $\displaystyle dx/dt=-3x/(1 plus t)^2$ (sorry no plus sign on this laptop :P )

Anyway, initially at rest, t=0, x= 1000. Show that at time T the volume is given by: x=1000e^(-3t/(1 plus t))

And then prove that the volume of water tends to a limit as t tends to infinite and find that limit to the nearest litre.

Any help would be VERY appreciated coz I am so stumped. Thank you!!!!

2. Originally Posted by Yehia
The volume of water in a solution during a chemical process varies with time and satisfies the equation: $\displaystyle dx/dt=-3x/(1 plus t)^2$ (sorry no plus sign on this laptop :P )

Anyway, initially at rest, t=0, x= 1000. Show that at time T the volume is given by: x=1000e^(-3t/(1 plus t))

And then prove that the volume of water tends to a limit as t tends to infinite and find that limit to the nearest litre.

Any help would be VERY appreciated coz I am so stumped. Thank you!!!!
$\displaystyle \frac{dx}{dt} = \frac{-3x}{(1+t)^2}$

$\displaystyle \frac{dx}{x} = \frac{-3}{(1+t)^2} \, dt$

$\displaystyle \ln|x| = \frac{3}{1+t} + C$

$\displaystyle x = Ae^{\frac{3}{1+t}}$

$\displaystyle 1000 = Ae^{3}$

$\displaystyle A = 1000e^{-3}$

$\displaystyle x = 1000e^{-3} \cdot e^{\frac{3}{1+t}}$

$\displaystyle x = 1000e^{-3 + \frac{3}{1+t}}$

$\displaystyle x = 1000e^{\frac{-3t}{1+t}}$

3. Originally Posted by Yehia
The volume of water in a solution during a chemical process varies with time and satisfies the equation: $\displaystyle dx/dt=-3x/(1 plus t)^2$ (sorry no plus sign on this laptop :P )

Anyway, initially at rest, t=0, x= 1000. Show that at time T the volume is given by: x=1000e^(-3t/(1 plus t))

And then prove that the volume of water tends to a limit as t tends to infinite and find that limit to the nearest litre.

Any help would be VERY appreciated coz I am so stumped. Thank you!!!!
$\displaystyle \frac{dx}{dt} = \frac{3x}{(1+t)^2}$

I'd go with separation of variables here

$\displaystyle \frac{dx}{3x} = \frac{dt}{(1+t)^2}$

Integrating both sides

$\displaystyle \frac{1}{3}\ln|x| = -\frac{1}{1+t} + C$

$\displaystyle ln|x| = -\frac{3}{1+t} + C$

$\displaystyle x = e^{-\frac{3}{1+t} + C} = C'e^{-\frac{3}{1+t}}$

where $\displaystyle C' = e^C$