Initial Value Problems

The Initial Value Problems (IVPs) that we will study are essentially just antiderivatives with an initial condition applied, which allows us to obtain the value of "", the constant of integration. Until we are given an initial condition (a point on the solution curve), we cannot determine the value of , and so the general solution is really an infinitely-large family of curves, one curve for each value of .
We write the equation as a Differential Equation (using a derivative): We then take the antiderivative of both sides, combining the two constants into one on the right side: This gives us the general solution. Finally, we apply the initial condition to obtain "", giving us the particular solution.