Ordinary Differential Equations
Introduction: In this section a brief summary of ordinary differential equations is presented. It is not intent to be a replacement to a standard textbook but as a quick reference. It is suggested that the reader for more in depth information will read “Differential Equations and Boundary Value Problems” by Boyce de–Prima or any other book in this area. Ordinary differential equations are defined by their order which is defined by the order of the highest derivative. If the highest derivative is first order the equation is referred as first order differential equation etc. Ordinary separated to linear and non-linear equations. The linearly is determined by the multiplication of function or its derivatives.
First Order Differential Equations: A first order derivative equation has several forms and there is no one solution fit all but families of solutions. The most general form is
Or it can be simplified to the first form as
In some case F(x, y) can be written as X(x)Y (y)in that case it is said that F is spreadable. Then equation (A.2) can be written as
Equation can be integrated either analytically or numerically and the solution is
The limits of the integral are (are) the initial condition. The initial condition is the value the function has at some point. The initial condition referred to the fact that the value are given commonly at time equal to zero.
The Integral Factor: Another method is referred to as integration factor which deals with a limited but very important class of equations. This family is referred as linear equation. The general form of the equation is
Multiplying equation by unknown function N(x) transformed it to
What is needed from N(x) is to provide a full differential such as
This condition requires that
Equation is integrated to be
So Thus equation becomes
The solution is then