## Solving Second Order Constant Coefficient Linear Differential Equations

The general second order differential equation with constant coefficients takes the form

We solve this in two parts.

First we solve the associated homogeneous equation

using the substitutionobtaining the characteristic equationWe can cancel the common factorto obtainThe roots of this equation may be real or complex.

If they are real and distinct,andthen the solution to the homogeneous equation is called the complementary solutionwhereandare arbitrary constants.

If there is only one real rootthen

If the roots are complex,and then

The following general solution can usually be quoted, where again A and B are arbitrary constants.

Next we look for a solutionof the full equation by choosing a function which 'looks' like

Ifchoose

Ifthen choose

Ifthen chooseetc.

This function is substituted into the original equation to be solved forand the coefficientsand orfound.

Then the general solutionis found by addingandFinally we find the coefficientsthat appear in the complementary solutionThese are found by substituting boundary conditions intoThese boundary condition consist of simultaneous values ofand If the equation to be solved is in terms ofandorandthese boundary conditions are called initial conditions.