In general, a linear n^th order differential equation can be written as

where the p_i(x) coefficients and right hand side forcing function, r(x), are continuous functions. For example, the standard form for a second order system is

where y(x) is the general solution, y_h(x) is the homogeneous solution containing n arbitrary constants, and y_p(x) is the particular solution containing no arbitrary constants.

We will discuss various analytical solution methods for linear ODEs and other related topics in this section of notes. Some of the topics highlighted, and their inter-relationships, are as follows:

Homogeneous Equations:

Particular Solutions:

A general constant coefficient homogeneous linear ODE can be written as

One usually assumes a solution of the form (for constant ). Substitution of this expression into the original equation gives

Dividing by gives the characteristic equation

and the roots of the characteristic polynomial represent values of that satisfy the assumed form for z(x). For an n^th order system, there will be n roots (not necessarily distinct) to the n^th order characteristic equation.

The general solution becomes a linear combination of the individual solutions to the homogeneous equation with the restriction that the n solutions must be linearly independent. For n distinct roots, the homogeneous solution can be written as

General Discussion

The subject of linear independence is very important since n linearly independent solutions are needed to form the homogeneous solution to an n^th order linear ODE. Here we will restrict our arguments to a second order system and then generalize for the n^th order case.

Let’s define y₁(x) and y₂(x) as two solutions to a 2^nd order linear ODE. Simply stated, if , then these two functions are linearly dependent. However, if , then y₁(x) and y₂(x) are linearly independent, since their ratio varies with x.

This check on linear independence is often quite straightforward for 2^nd order systems, but for the general case, simply looking at ratios of functions is usually not sufficient. However, this check can be generalized as shown below.

General Check for Linear Independence

Second Order Systems:

The homogeneous solution is a linear combination of the two linearly independent individual solutions, y₁(x) and y₂(x), or

Now, if y₁(x) and y₂(x) are really linearly independent, then this is a valid homogeneous solution. As such, we can take the derivative of this expression, giving

Writing these two equations, evaluated at any value on the interval of interest, as a matrix equation gives

For this (assumed to be valid) system to have a non-trivial solution, the coefficient matrix must not be singular (i.e. its determinant must be non-zero). Thus we have the condition that

implies linear independence.

This particular determinant appears frequently and is called the Wronskian of y₁ and y₂ and is denoted as W(y₁, y₂). Therefore,

implies that y₁ and y₂ are linearly independent.

Higher Order Systems:

The above result extends nicely to n^th order systems. In particular,

implies that are all linearly independent.

General Discussion

One often needs to obtain a second solution when one solution is already known. This occurs, for example, if the two individual solutions to a 2^nd order system are linearly dependent, or if one solution is known by inspection of the equations (or by some other means). In these cases, the Reduction of Order Method can be used to final a second linearly independent solution.

Development of the Method

Given a valid solution to a homogeneous linear 2^nd order differential equation, one can obtain a second linearly independent solution by letting

where y₁(x) is known and u(x) is to be determined. This technique applied to a 2^nd order system is as follows:

Given the original differential equation

with known solution y₁(x), we let

or

Now, substitution into the original equation gives

or

Letting z = u’ and noting that the third term vanishes from definition of the original ODE, we have

Now, separating variables and integrating gives

or

Finally, since du/dx = z, we have

General Note

If, in a 2^nd order equation, the dependent variable y does not appear explicitly, one has . In this case, one can let z = y’ and obtain a 1^st order equation for z and then find y(x) by integration. This technique was used in the above development, and it is quite useful in many situations.