### Unlimited growth: Exponential growth

### The differential equation of exponential growth

As noted before, for instance at 'linear growth', the differential equation alone does not have a single solution, but a whole family of solutions. To choose a solution from this family, other conditions have to be given. For instance, when the value at a certain moment \(t\) in time is known (often \(t=0\)), the particular solution out of the family of solutions can be chosen. A differential equation and such a specified value together are called an **initial value problem.** Similarly, a differential equation and other contraints together are called a **boundary value problem.**

Below are a few examples of the differential equation of exponential growth.

This is a differential equation of exponential growth with relative growth rate \(-1\).

Thus, the general solution is the same as \[y(t)=c\cdot e^{-t}\] Substituting the condition \(y(0)=9\) leads to the equation \[c\cdot e^{0} = 9\] This can be simplified to: \[c=9\] The solution of the initial value problem is \[y(t)=9\,e^{-t}\]

Thus, the general solution is the same as \[y(t)=c\cdot e^{-t}\] Substituting the condition \(y(0)=9\) leads to the equation \[c\cdot e^{0} = 9\] This can be simplified to: \[c=9\] The solution of the initial value problem is \[y(t)=9\,e^{-t}\]