## Solving Second Order Constant Coefficient Linear Differential Equations

A second order constant coefficient linear differential equation is any equation of the form where and are constants. Solving these involves

1. Finding a solution of the homogeneous equation We may assume a solution of the form obtaining the 'characteristic' equation If this equation has two distinct real roots and then If the equation has a single repeated root then If the equation has complex roots then 2. Now find a particular solution of the inhomogeneous equation by assuming a solution similar to the function For example if is a polynomial of degree then assume is also a polynomial of degree so that if then assume The constants can be found by equating coefficients.

3. The general solution is the sum of the complementary and particular solutions: The constants and can be found if we are given two conditions on and Example: Solve the equation if and at Assuming a solution to the homogeneous equation of the form we have and The equation becomes The complementary solution is then Assume a particular solution of the form and so the equation becomes Equation coefficients of gives Equating constants gives The particular solution is then We have The first condition gives  The second condition gives The solution to the problem is  