2.5.4 Differential-Equation-with-Initial-Value
         Problems

In general, there are many solutions of a differential equation such as

infinitely many, in fact. However, it is certainly plausible that if we choose an initial point $(t_0,P_0)$, and if we know the slope of the solution at every point $(t,P)$, then there must be only one solution that passes through $(t_0,P_0)$. Thus, in hope of finding a uniquely determined solution — and because it's reasonable to assume we know a starting population — we turn our attention to a differential-equation-with-initial-value problem:

Such a problem describes a unique function $P=P\left(t\right)$.

For the differential-equation-with-initial-value problem

we have already done most of the work for finding a formula for the solution. Indeed, if we let $P=A{e}^{0.25t}$, then we know from Section 2.4 that

Thus, for every constant \(A\), the function $P=A{e}^{0.25t}$ is a solution of the differential equation

Now we choose \(A\) so that $P=1000$ when $t=0$:

Thus $P=1000e^{0.25t}$ is the desired solution.

Activity 1

Show that the general differential-equation-with-initial-value problem

has the solution

Comment on Activity 1