3.1.2 Initial Values

The slope fields in Figures 1, 2, and C1 suggest that, for each point in the $(t,y)$-plane, there is exactly one solution curve passing through that point. Our solution formulas suggest the same thing: One $(t,y)$ pair of numbers that must satisfy the formula is enough information to determine the constant `C` in each case. Thus, in order to specify a particular solution of the differential equation, all we have to do is specify its value at one point.

Activity 4

1. Find the particular solution of the differential equation $\frac{dP}{dt}=0.25P$ that satisfies the condition $P\left(0\right)=2$.

2. Find the particular solution of the same differential equation that satisfies the condition $P\left(1\right)=2$.

3. Use a graphing tool to graph your solution functions from parts (a) and (b) on the slope field for the differential equation.

4. Confirm that each of your solutions does indeed satisfy the corresponding condition stated in part (a) or (b).

Comment on Activity 4

The number "2" that appears in parts (a) and (b) of Activity 4 is called an "initial value." The statement that `P` must have the value `2` at $t=0$ (or at $t=1$) is called an "initial condition." The entire problem stated in either part (a) or part (b) of Activity 4 is called an "initial value problem." We now give formal definitions of these terms.

Definition   An initial value is a specified value of a particular solution function at a specific number in the domain of the function. An initial condition is the specification of an initial value for a solution function. An initial value problem consists of a differential equation together with an initial condition. A solution of an initial value problem is a solution of the differential equation that also satisfies the initial condition.