Non-homogeneous second-order equations

Special case of the general linear equation of order n

(1)

where and are given functions of or are constants.

Consider equation (1). If , it is called homogeneous equation with variable or constant coefficients, depending on whether are functions of or are constants. It is also called the reduced linear differential equation.

Example 10. ,is a second order homogeneous equation with variable coefficient.

If in equation (1), it is called non- homogeneous equation with variable or constant coefficients, depending on whether are functions of or are constants. Example 11. is a second order non-homogeneous equation with variable coefficient.

Suppose and are two independent solutions of the reduced equation of (1), namely

(2)

then the linear combination where are arbitrary constants, is also a solution.

Theorem 1. If are n linearly independent functions of which satisfy a homogeneous equation (3), then the linear combination

(3)

where are arbitrary constants, is its solution. Equation (3), which provides the solution for the homogeneous equation is called the complementary function.

Theorem 2. The general solution of a complete non-homogeneous differential equation is equal to the sum of its complementary function and any particular integral. If is a particular solution of (1), then the general solution is

+ (4)

Hence, for non-homogeneous equations:

The general linear equation (4) is usually difficult to solve and requires special techniques. However an important and special case occurs when the coefficients are constants, the equation being called a constant coefficient equation. Consider a constant coefficient homogeneous equation

(5)

Denoting, , in equation (5) we have

. (6)

If we make a formal substitution in (6), we obtain a polynomial in

of degree given:

, (7)

and if we equate equation (7) to zero we have an algebraic equation of degree which must have roots. The equation is called the auxiliary equation of the differential equation of the differential equation (7).

Theorem 3. If is a root of the auxiliary equation , then is a solution of the homogeneous linear differential equation

where is a constant.

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