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.
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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.
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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.
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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.
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This example has roots with nonzero imaginary parts.
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Here is a plot of the three solutions.
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