This is documentation for Mathematica 6, which was
based on an earlier version of the Wolfram Language.
 Mathematica Tutorial

# Linear Second-Order Equations with Constant Coefficients

The simplest type of linear second-order ODE is one with constant coefficients.
This linear second-order ODE has constant coefficients.
 Out[1]=
Notice that the general solution is a linear combination of two exponential functions. The arbitrary constants C[1] and C[2] can be varied to produce particular solutions.
This is one particular solution to the equation.
 Out[2]=
The exponents -6 and 1 in the basis {-6x, x} are obtained by solving the associated quadratic equation. This quadratic equation is called the auxiliary or characteristic equation.
This solves the auxiliary equation.
 Out[3]=
The roots are real and distinct in this case. There are two other cases of interest: real and equal roots, and imaginary roots.
This example has real and equal roots.
 Out[4]=
 Out[5]=
This example has roots with nonzero imaginary parts.
 Out[6]=
 Out[7]=
Here is a plot of the three solutions.
 Out[8]=