Linear differential equation

In mathematics, a linear differential equation is a differential equation

Lf = g,

where the differential operator L is a linear operator. The condition on L rules out operations such as taking the square of the derivative of f; but permits, for example, taking the second derivative of f. Therefore a fairly general form of such an equation would be

where D is the differential operator d/dx, and the a_i are given functions. Such an equation is said to have order n, the index of the highest derivative of f that is involved.

The case where g = 0 is called a homogeneous equation, and is particularly important to the solution of the general case (by a method traditionally called particular integral and complementary function. When the a_i are numbers, the equation is said to have ''constant coefficients.

Table of contents

1 Homogeneous linear differential equation with constant coefficients
2 Inhomogeneous linear differential equation with constant coefficients
3 Other meanings

Homogeneous linear differential equation with constant coefficients

To solve such an equation one makes a substitution

y=e^λx,

to form the characteristic equation

to obtain the solutions

When this polynomial has distinct roots, we have immediately n solutions to the differential equation in the form

Then the general solution to the homogeneous equation can be formed from a linear combination of the y_i, ie.,

Where the solutions are not distinct, it may be necessary to multiply them by some power of x to obtain linear independence; the general solution therefore involves the product of polynomials and exponentials.

Inhomogeneous linear differential equation with constant coefficients

To obtain the solution to the inhomogeneous equation, find a particular solution y_P(x) by the method of undetermined coefficients; the general solution to the linear differential equation is the sum of the homogeneous and the particular solution.

Other meanings

A linear differential equation can also refer to an equation in the form

where this equation can be solved by forming the integrating factor , multiplying throughout to obtain

which simplifies due to the product rule to

on integrating both sides yields