## Solving Second Order Constant Coefficient Homogeneous

Second order differential equations arise naturally as a result of applying Newton's second law of motion. If no external forces act then the equation may take the form where is a drag or resistance term and is a restoring force.

If there is no resistance term then the equation becomes - the equation of simple harmonic motion - and this equation has the solution If there is no restoring force the the equation becomes We can write this equations as Integration gives We can solve this equation by separation of variables Integration gives and solving for x gives If there are both resistance and restoring terms then there are three possibilities.

We can substitute so that and The equation becomes We can divide by (since an exponential is never zero) to obtain This is a quadratic equation with roots If then there are two distinct real roots and and If then there is one root and In both of these cases, may increase without limit if is positive, and the motion is non – oscillatory.

If then there are two distinct complex roots and and This last solution for represents oscillatory motion. If the oscillations decay and if the oscillations increase in size.