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 \(y(0)=-5\) leads to the equation \[c\cdot e^{0} = -5\] This can be simplified to: \[c=-5\] The solution of the initial value problem is \[y(t)=-5\,e^{t}\]
Thus, the general solution is the same as \[y(t)=c\cdot e^{t}\] Substituting the condition \(y(0)=-5\) leads to the equation \[c\cdot e^{0} = -5\] This can be simplified to: \[c=-5\] The solution of the initial value problem is \[y(t)=-5\,e^{t}\]
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