Section Summary

In this section we have seen that a differential equation can specify slope information about an unknown function either in terms of the dependent variable or in terms of the independent variable. Either way, such an equation will have an infinite number of solutions. With the specification of an initial condition, the number of possible solutions is reduced to one. We can solve initial value problems in a two-step process: First, find the entire family of solutions of the differential equation, which will involve an arbitrary (or not-yet-specified) constant in the formula. Then substitute the initial condition to determine the value of the arbitrary constant.