differential equations of the form

(1) y⁽ⁿ⁾ + p₁(x)y^{(n - 1)} + ... + p_n(x)y = f(x)

where y = y(x) is an unknown function; y⁽ⁿ⁾, y^{(n - 1)}, . . ., y are its derivatives; and p₁(x), p₂(x), . . ., p_n (x) (the coefficients) and/(x) (the free term) are given functions. Equation (1) is said to be linear because the unknown function y and its derivatives enter (1) linearly, that is, the degree of y and its derivatives in (1) is one. If f(x) ≡ 0, then equation (1) is said to be homogeneous; otherwise it is inhomogeneous. The general solution y₀ = y₀(x) of a homogeneous linear differential equation with continuous coefficients p_k(x) can be written as

y₀ = C₁y₁(x) + C₂y₂(x) + ... + C_ny_n(x)

where C₁, C₂, . . ., C_n are arbitrary constants and y₁(x), y₂(x), . . ., y_n(x) are linearly independent particular solutions. Such n solutions are said to form a fundamental system of solutions. An important result states that n particular solutions are linearly independent if and only if their Wronskian

is different from zero at at least one point.

The general solution y = y(x) of the inhomogeneous linear differential equation (1) has the form

y = y₀ + Y

where y₀ = y₀(x) is the general solution of the corresponding homogeneous linear differential equation and Y = Y(x) is a particular solution of the given inhomogeneous linear differential equation. The function Y(x) can be found from the formula

where the y_k(x) form a fundamental system of solutions of the homogeneous linear differential equation and W_k(x) is the cofactor of the element y_k^{(n - 1)}(x) in the Wronskian w(x) given above in (2).

If the coefficients of equation (1) are constant, that is, p_k(x) = a_k for k = 1, 2, . . ., n, then the general solution of the homogeneous equation is given by

Here, ) are the roots of the so-called characteristic equation

λⁿ + a₁ λ^{n - 1} + ... + a_n = 0

and the n_k are the multiplicities of these roots and C_ks and D_ks are arbitrary constants.

Example. For the linear differential equation y″′ + y = 0, the characteristic equation has the form λ³ + 1 = 0. Its roots are

Consequently, the general solution of this equation is

Consider a system of n linear differential equations

The general solution of the homogeneous system of linear differential equations [obtained from (3) by putting all f_j(x) ≡ 0] is given by

where y_j1, y_{j 2}, . . ., y_jn are linearly independent particular solutions of the homogeneous system (that is, solutions such that the determinant ǀy_jk (x)ǀ≠ 0 at at least one point).

In the case of constant coefficients p_jk(x) = a_jk, particular solutions of the homogeneous system are of the form

where the A_js are undetermined coefficients and the λ_k are the roots of the characteristic equation

with multipliers m_k. Here, a complete analysis of all possible cases can be effected by means of the theory of elementary divisors.

The methods of operational calculus are also used to solve linear differential equations and systems of linear differential equations with constant coefficients.