Solving Second Order Constant Coefficient Linear Differential Equations

A second order constant coefficient linear differential equation is any equation of the form

whereandare constants. Solving these involves

  1. 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

  2. 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.

  3. The general solution is the sum of the complementary and particular solutions: The constantsandcan be found if we are given two conditions onand

Example: Solve the equationifandat

Assuming a solution to the homogeneous equationof the formwe 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