1. ## differential equation

A neglected large lawn contains a certain type of weed. The area of the lawn covered by the weed at time t years is Wm². The rate of increase of W is directly proportional to W.

Write down a differential equation that is satisfied by W.

OK now i know the answer is [dw/dt = KW] because these questions go by the same format all the time

BUT

I need to know WHY it is that. I don't want any nasty surprises when i go into my exam :/

2. Originally Posted by djmccabie
A neglected large lawn contains a certain type of weed. The area of the lawn covered by the weed at time t years is Wm². The rate of increase of W is directly proportional to W.

Write down a differential equation that is satisfied by W.

OK now i know the answer is [dw/dt = KW] because these questions go by the same format all the time

BUT

I need to know WHY it is that. I don't want any nasty surprises when i go into my exam :/
If one variable is directly proportional to another variable, it means that they are constant multiples of each other. So in this case, one variable is $\frac{dW}{dt}$ and the other variable is $W$, so you have the equation $\frac{dW}{dt}=kW$, where $k$ is a constant.

Separate the variables and integrate: $\int\frac{dW}{W} = \int k\,dt \implies \ln(W) = kt+C \implies W=e^{kt+C} \implies W=c_0e^{kt}$ (where $c_0=e^C$).