Second order linear ODE

Second order linear ODEs are in the following form:

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Important types of second order linear ODEs include

• Homogeneous
• Constant coefficients (where p and q are constants)

Initial value problem

There are two arbitrary constants in the solution of a second order linear ODE, so we need two initial conditions.

${\begin{cases}y(t_{0})=y_{0}\\y'(t_{0})=y_{0}'\end{cases}}$

Solutions

Constant coefficient, homogeneous

These are the simplest kind. They have the general form

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An exponential function has the property of being the same after many differentiation. We take advantage of this property and guess the solution to be the form of

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