In this article, a numerical technique called shooting which entails the solution of initial value problems with the single-step fourth-order Runge-Kutta method together with the iterative root finding secant method is formulated for use on both linear and non-linear boundary value problems of ordinary differential equations with Dirichlet boundary conditions. Two examples are illustrated. One, the solution of the linear case with its analytic counterpart is compared, and two, the non-linear case. Graphical outputs of the solutions from two MATHEMATICA codes are presented.

Differential equations express the relationship between unknown function or functions and their derivatives. Usually, the unknown functions represent physical quantities (heat transfer in a medium, flow of a current in the electric circuit, the evolution of price options in mathematical finance, etc.,) and their derivatives represent their rates of change. This is one reason why we endeavor to study differential equations, they are the tools used in the world of mathematical modeling. The study of differential equations concentrates mainly on their solutions and the properties of these solutions. When a solution of a differential equation cannot be found by explicit formulae, the solution may be approximated by numerical methods.

The classification of differential equations is important because it helps to choose the type of solution method required for a particular differential equation. Classification is based on some of the many concepts, namely; ordinary or partial differential equations, linear or non-linear equations, homogeneous or heterogeneous equations, order of differential equations, and others.

A differential equation whose unknown function depends only on one independent variable is called an ordinary differential equation. A differential equation whose unknown function depends on more than one independent variable is called a partial differential equation.

In this article, our attention will be focused on ordinary differential equations.

Ordinary Differential Equation is in most cases abbreviated as ODE. The unknown function is called the dependent variable. If

the dependent variable

all the coefficients

If any of these two conditions is not satisfied then the ODE is nonlinear. Usually, explicit solutions of nonlinear ODEs are difficult to find.

Explicit formulae solutions of ordinary differential equations are limited. At best there are only a few ordinary differential equations that can be solved analytically. There are many different methods in the literature today for analytical solutions of both IVPs and BVPs [

Solution of the nonlinear equation

Graphical derivation of this method is left to the reader. Note that this method requires two initial guess values

Hence the root is

Let us now relate the above information to the shooting method. This is a numerical method that can be used to find solutions of BVPs of ordinary differential equations. Given a BVP of the ordinary differential equation, first reduce it to a system of first order equations. The key task here is to solve the above equations as IVPs. One or more of the resulting IVPs (depending on the order of the BVP) will not have an initial value at the first boundary point. Let the value of this IVP at the first boundary point be say

The solution of the IVP at the second boundary point say

In this section, we shall have two examples. The first one for a simple linear BVP whose analytic solution is readily obtained and the second example is a non-linear BVP whose solution is not readily obtained by analytical means. Each example is accompanied by a MATHEMATICA code for the solution.

If we choose (guess) a value for the solution

The solution is shown in

The solution of the linear BVP using the shooting technique matches very well with its analytic solution. This motivated us to find the solution of the non-linear BVP. Solution techniques are numerous in the literature. The shooting method formulated in this article is one method that can be used to solve linear as well as non-linear boundary value problems of ordinary differential equations with Dirichlet boundary conditions.