# 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[1] and C[2] 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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