General Discussion

A 2^nd order constant coefficient homogeneous system can be written as

The corresponding characteristic equation is

and the roots of this quadratic equation are given by

With known (distinct) roots, the general solution can be written as

The actual form of the solution is strongly dependent on whether the roots are real versus complex or distinct versus repeated. In fact three special cases can be identified based on whether the term inside the radical is positive, negative, or zero. These three cases are identified in detail in the remainder of this subsection.

Three Special Cases

I. Real Distinct Roots:

One particular subset here is when a = 0 and b < 0. For this special case, and the solution, y(x), may also be written as

The correspondence between the coefficients, c₁, c₂ and k₁, k₂, for the two forms for this special case is easily developed if one recalls the definition of the hyperbolic functions in terms of the real exponentials, or

Now putting these into eqn. (2.21) gives

and from the last relationship, it is obvious that the correspondence among the coefficients is given by

II. Complex Conjugate Roots:

Recalling the definition of the complex exponential and the complex form of the sine and cosine functions,

the standard form of the solution, y(x), can be rewritten using

to get

where now the relationship between the two formulations is

III. Repeated Roots:

Therefore, there is only one independent solution,

we have from eqns. (2.16) and (2.17) that

For the present case, p(x) = a and , therefore z(x) becomes

and u(x) is given by

Therefore,

Special Note

This technique of multiplying by some power of x is quite useful for Linear Constant Coefficient Equations with repeated roots. This is also true for particular solutions to constant coefficient ODEs when the forcing function is of the same form as the homogeneous solution (to be discussed later).

General Discussion

In general, variable coefficient linear ODEs are rather difficult to solve analytically. In a subsequent section we will use the Power Series Method to address this problem from an analytical viewpoint, and we will use a numerical scheme for the general case. However, there is one special case of a variable coefficient system, called the Euler-Cauchy equation, that occurs quite frequently in applications and has a solution scheme very similar to that of constant coefficient systems. This subsection is devoted to developing the solution method for this special system - the homogeneous 2^nd order Euler-Cauchy equation.

For 2^nd order, the standard Euler-Cauchy equation has the form

where a and b are constants. We assume a solution of the form

Therefore,

Upon substitution into the original ODE given in eqn. (2.33), we get

and dividing by x^m gives the characteristic equation,

The roots of this quadratic equation are given by

As for the case of constant coefficients, the actual form of the solution is strongly dependent on whether the roots are real versus complex or distinct versus repeated. In fact three special cases can be identified based on whether the term inside the radical is positive, negative, or zero. These three cases, for the Euler Cauchy equation, are identified in detail in the remainder of this subsection.

Three Special Cases

I. Real Distinct Roots:

II. Complex Conjugate Roots:

Before writing the general solution in terms of complex roots, however, let’s look at alternate ways to write x^m. The goal here is to write the solution in terms of real functions, if possible. First consider the following form for x^m,

Now, if , then

and similarly

Therefore,

or

where

Note also that the following expressions are often useful in justifying the relationship of the standard solution in eqn. (2.37) and the form given in eqn. (2.42):

III. Repeated Roots:

Therefore, there is only one independent solution,

we have from eqns. (2.16) and (2.17) that

For the present case, p(x) = a/x , q(x) = b/x², and . Therefore z(x) becomes

but m = (1-a)/2, thus , and

Finally,

Therefore,

Special Note

This technique of multiplying by ln(x) is quite useful for Euler-Cauchy equations with repeated roots. This is also true for particular solutions when the forcing function for the Euler-Cauchy equation is of the same form as the homogeneous solution (to be discussed shortly).