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.
Notice that the general solution is a linear combination of two exponential functions. The arbitrary constants C and C can be varied to produce particular solutions.
This is one particular solution to the equation.
The exponents and in the basis are obtained by solving the associated quadratic equation. This quadratic equation is called the auxiliary or characteristic equation.
This solves the auxiliary equation.
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.
This example has roots with nonzero imaginary parts.
Here is a plot of the three solutions.