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

Finding a solutionof the homogeneous equationWe may assume a solution of the formobtaining the ‘characteristic’ equationIf this equation has two distinct real rootsandthenIf the equation has a single repeated rootthenIf the equation has complex rootsthen

Now find a particular solutionof the inhomogeneous equationby assuming a solution similar to the functionFor example ifis a polynomial of degreethen assumeis also a polynomial of degreeso that ifthen assumeThe constantscan be found by equating coefficients.

The general solution is the sum of the complementary and particular solutions: The constantsandcan be found if we are given two conditions on and
Example: Solve the equationifandat
Assuming a solution to the homogeneous equationof the form we haveandThe equation becomes
The complementary solution is then
Assume a particular solution of the formandso the equation becomes
Equation coefficients ofgivesEquating constants givesThe particular solution is thenWe have
The first conditiongives
The second condition gives
The solution to the problem is