Linear differential equation

Definition

In explicit form

A linear differential equation of order $k$ looks as follows in explicit form:

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

General version

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.
homogeneous 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
second-order homogeneous case, found one solution already	Another solution can be obtained by solving a first-order linear differential equation involving the first solution, and then the general solution is obtained by allowing arbitrary linear combinations.
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.
homogeneous case	If the equation has order $k$ and we find $k$ linearly independent solutions, then the solution space is precisely the vector space generated by these solutions. Although there is no procedure for finding these solutions in general, this fact does tell us that if we are somehow able to find the solutions, we'll know there are no more.