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 "[math]C[/math]", the constant of integration.[br][br]Until we are given an initial condition (a point on the solution curve), we cannot determine the value of [math]C[/math], and so the general solution is really an infinitely-large family of curves, one curve for each value of [math]C[/math].

We write the equation as a Differential Equation (using a derivative):[br][br][math]y'=f(x)[/math][br][br]We then take the antiderivative of both sides, combining the two constants into one on the right side:[br][br][math]\int y'dx=\int f\left(x\right)dx[/math][br][math]y=F\left(x\right)+C[/math][br][br]This gives us the [b][i]general solution[/i][/b]. Finally, we apply the initial condition to obtain "[math]C[/math]", giving us the [b][i]particular solution[/i][/b].