## 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 substitution obtaining the characteristic equation We can cancel the common factor to obtain The roots of this equation may be real or complex.

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

If there is only one real root then 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 solution of the full equation by choosing a function which 'looks' like If choose If then choose If then choose etc.

This function is substituted into the original equation to be solved for and the coefficients and or found.

Then the general solution is found by adding and Finally we find the coefficients that appear in the complementary solution These are found by substituting boundary conditions into These boundary condition consist of simultaneous values of and If the equation to be solved is in terms of and or and these boundary conditions are called initial conditions. 