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
whereis a drag or resistance term andis 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 givesWe can solve this equation by separation of variables
Integration givesand solving for x gives
If there are both resistance and restoring terms then there are three possibilities.
We can substituteso thatandThe equation becomes
We can divide by(since an exponential is never zero) to obtain
This is a quadratic equation with roots
Ifthen there are two distinct real rootsandand
Ifthen there is one rootand
In both of these cases,may increase without limit ifis positive, and the motion is non – oscillatory.
Ifthen there are two distinct complex roots
This last solution forrepresents oscillatory motion. Ifthe oscillations decay and if the oscillations increase in size.