Second Order Differential Equation

Second order differential equations also arise frequently in applications. As a rule, the general solution of a second order differential equation will involve two constants, and two initial conditions are needed to determine a particular solution.

We shall now discuss an important second order differential equation whose solution involves sines and cosines.

The general solution of the equation

is                                                y = a cos t + b sin t.

We have

Therefore both y = sin t and y = cos t are solutions. It then follows easily that every function a cos t + b sin t is a solution. Notice also that if

y = a cos t + b sin t then at time t = 0, y = a and dy/dt = b.

It can be proved that there are no other solutions, but we shall not give the proof here.

More generally, given a constant ω the equation

has the general solution

y = a cos cot + b sin cot.