Linear differential equation

Definition

In normalized form

A linear differential equation of order $k$, normalized for $y^{(k)}$, is of the form:

$y^{(k)}+p_{k-1}(x)y^{(k-1)}+p_{k-2}(x)y^{(k-2)}+\dots +p_{0}(x)y=q(x)$

The general version allows for a coefficient of $y^{(k)}$ that is a function of $x$. It can be converted to the above form by dividing throughout by that coefficient.

Particular cases

• Linear differential equation with constant coefficients refers to a case of a linear differential equation where all the functions $p_{k-1},p_{k-2}\dots ,p_{0}$ are constants.
• Homogeneous linear differential equation refers to a case where $q(x)$ is the zero function.
• Homogeneous linear differential equation with constant coefficients refers to a case where all the functions $p_{k-1},p_{k-2}\dots ,p_{0}$ are constants and $q(x)$ is a constant function. There is a general procedure for solving such equations.

Solution

There is no known general procedure for solving linear differential equations. However, the following are known:

Shorthand for solution or reduction method	Details
first-order case	If the order of the differential equation is one, it is known how to convert it to an integration problem, i.e., there is a known integrating factor that will make the differential equation a first-order exact differential equation. For more, see first-order linear differential equation.
reduction to homogeneous case	Solving a linear differential equation is equivalent to (solving the corresponding homogeneous linear differential equation (obtained by replacing $q(x)$ with zero) + finding a particular solution). In fact, the general solution to a linear differential equation is of the form (any particular solution) + (general solution for corresponding homogeneous linear differential equation). Thus, the problem of solving a linear differential equation can be neatly split into two parts.
homomgeneous constant coefficients case	There is an explicit description of the general solution in terms of the coefficients. It involves finding the roots of a polynomial whose coefficients are the coefficients of the differential equation. For more, see homogeneous linear differential equation with constant coefficients