Chapter 8. Solving Differential Equations Numerically We can think of solving differential equations as the inverse of finding the derivative of a function. Given a differential equation (the derivative), a solution is any function that has that derivative. In this chapter, we will deal with only first-order ordinary differential equations (that is, first derivatives only). For example, a solution of the differential equation [1] [1] Differential equation problems traditionally use y and y ' to name a function and its derivative. 
is 
where C is an arbitrary constant. The latter is a solution because its derivative is the differential equation. Because of the arbitrary constant C, there are an infinite number of solutions. In order to have a particular solution, we need to add an initial condition. Therefore, if we modify the problem in the previous paragraph to finding a solution to the differential equation with the initial condition y (0) = -5, then we have an initial value problem, and its solution is 
Like integration problems in freshman calculus, solving initial value problems is straightforward only if the differential equations fit certain well-known patterns. Very often, they don't, and so we resort to using computers to generate numerical solutions. As you may have guessed, a numerical solution of an initial value problem is only an approximation to the true, analytical solution. Also, we won't get a solution written out symbolically, like the one shown earlier. A numerical solution generally is a set of points that lie on the approximation. Our goal is to devise algorithms that generate the most accurate solutions in the least amount of time. In this chapter, we'll investigate three algorithms that solve initial value problems numerically.  | 
