Second-Order Ordinary Differential Equations
An ordinary differential equation (ODE) is an equality involving a function containing an unknown and the derivative(s) of that function. The order of an ordinary differential equation is determined by the order of the highest-order derivative of the function appearing in the equality. A second-order ordinary differential equation contains the second derivative of the function f(x, y) and is usually written as: d2y/dx2 + Bdy/dx = f(x, y), where d2y/dx2 is the second derivative of the function f with respect to x, and dy/dx is the first derivative of the function f with respect to x. A solution to a second-order ordinary differential equation is any function y that satisfies that differential equation. Second-order ordinary differential equations have two linearly independent solutions and any linear combinations of those linearly independent solutions are also solutions. Second-order ordinary differential equations have a wide variety of uses including calculating the changing size of a population, determining the flow of current in an electric circuit, describing the motion of a pendulum, and describing the motion of a weight attached to the end of a spring. These uses are of interest in chemistry, physics, economics, engineering, and electronics.
There are different classifications of second-order ordinary differential equations that help in determining which methods are most effective for solving them. For second-order differential equations there are two types of solutions: the particular solutions and the general or complementary solution. Second-order ordinary differential equations can also be expressed as a system of two first-order differential equations. This property is used to solve them. If a second-order ordinary differential equation has the form: d2y/dx2 + P(x)dy/dx + Q(x)y = G(x) where P(x), Q(x) and G(x) are continuous then it is said to be a linear second-order ordinary differential equation. If the equation is similar to the form above but has the form: d2y/dx2 + bdy/dx + cy = G(x) where b and c are constants and G(x) = 0 then it is called homogeneous as well as linear. This type of equation can be solved by employing its characteristic or auxiliary equation and determining the roots by the quadratic formula. The resulting solution will be the complementary solution. If G(x) 0 the linear second-order ordinary differential equation is considered nonhomogeneous and another method must be employed to find the solutions. The general solution of a nonhomogeneous linear second-order differential equation is the sum of a particular solution and the complementary solution. Since the complementary solution can be found as previously described it remains to find only the particular solution. The method of variation of parameters is employed to find the particular solution.
Some linear second-order ordinary differential equations have variable coefficients, that is P(x) and Q(x) from above are functions rather than constants. For certain equations the variable coefficients can be transformed into constant coefficients by substituting another function z that is equivalent to the functions acting as coefficients. These particular differential equations for which this can be done have: (Q'(x) + 2P(x)Q(x))/(2(Q(x)3/2)) = constant, where Q'(x) is the first derivative of the function Q(x) with respect to x.
There are other special classes of second-order ordinary differential equations that include x-missing, which is just an equation that does not involve a function of x and has the form: d2y/dx2 = f(y, dy/dx). Similarly there are y-missing equations that have the form: d2y/dx2 = f(x, dy/dx). Some linear second-order differential equations can be simplified by eliminating the first-order term via substitution so that it is transformed into standard form: d2z/dx2 + q(x)z = 0.
The solutions of second-order ordinary differential equations can be classified as to their singularity at the origin. A solution is said to be singular if at that point, the singular point, the function fails to be analytic. Linearly independent solutions to second-order ordinary differential equations that are singular at the origin are classified as "of the second kind". If the solution is nonsingular at the origin it is said to be "of the first kind".
Although there are many methods for solving different kinds of second-order ordinary differential equations the only practical solution method for very complex equations is to use numerical methods. All solutions to second-order ordinary differential equations satisfy existence and uniqueness properties. One last note concerning the solutions to second-order ordinary differential equations that makes determination of the general solution easier is that if one solution of a particular equation is known then the other solution can be found using the reduction of order method. This is because of the relationship between the particular solutions and the general solution.
This is the complete article, containing 756 words
(approx. 3 pages at 300 words per page).
