Principle of Second Order Superposition

Suppose x(t) and y(t) are solutions of the two second order linear differential equations

ax" + bx' + ex =f(t) ay" + by' + cy = g(t).

Then for any constants A and B, the function

u = Ax + By is a solution of the linear differential equation

au" + bu' +-cu = Af(t) + Bg(t).

Theorem 1 breaks the problem of finding the general solution of the equation (1) into two simpler problems.

First problem: Find the general solution of the corresponding linear homogeneous equation

ax" + bx' + c = 0.

Second problem: Find some particular solution of the given equation

ay" + by' + cy=f(t).

The first problem was solved in the preceding section. We now present a method for solving the second problem. This method is sometimes called the method of judicious guessing, or the method of undetermined coefficients. The method works only when the forcing term f(t) is a fairly simple function, of the form

(3)

 p(t)^αt cos (pt) + q(t)e^αt sin (βt),

where p(t) and q(t) are polynomials. However, when it works it is a very efficient method of solution. Often f(t) will be of an even simpler form, such as a polynomial alone, or a single exponential or trigonometric function. In the case of a homogeneous equation, where f(t) = 0, the zero function y(t) = 0 is a particular solution. The idea for solving a linear equation is to guess that the differential equation has a particular solution, which looks like the forcing term f(t) but has different constant coefficients. By working backwards, it is possible to find the unknown constants and discover a particular solution. We illustrate the method with several examples.

Example 1

In the remaining examples we shall concentrate on the first part of the problem, finding some particular solution of the given differential equation. In each case we could then find the general solution by solving the corresponding homogeneous equation and applying Theorem 1 as we did in Example 1.

Example 2

Example 3

Example 4

Example 5

Example 6