1D

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In one dimension we are mostly concerned with the homogenous 2nd order ODE, which has three cases, underdamped, critically damped, and overdamped. An underdamped example is,

which has the solution

which can be plotted as

A critically damped example is,

which has the solution

which can be plotted as

An overdamped example is

which has the solution

which can be plotted as

There are also nonhomogenous 2nd order ODEs which have more interesting behavior.

with initial conditions

has the solution