Solving Constant Coefficient, Linear, Second Order, Homogeneous Equations - Examples

Solving a constant coefficient equation – one of the form– may be accomplished by assuming a solution of the formThe solution will contain two arbitrary constants, which can be found given two boundary conditions onor some derivative of

Example: Solve the equationsubject to the conditionswhenandwhen

AssumingthenandSubstitute these into the original equation to getWe divide by the none zero factorto obtainwhich factorises to giveWe obtain the two solutions

The solution is thenWe can findandusing the conditions given.

(1)

(2)

(1)+(2) givesthen from (1)so

Example: Solve the equationsubject to the conditionswhenandwhen

AssumingthenandSubstitute these into the original equation to getWe divide by the none zero factorto obtainwhich factorises to giveWe obtain the solution (twice).

The solution is thenWe can findandusing the conditions given.

so the second condition

The solution is then

Example: Solve the equationsubject to the conditionswhenandwhen

AssumingthenandSubstitute these into the original equation to getWe divide by the none zero factorto obtainwhich has the solutionswhere

The solution is then

We can findandusing the conditions given.

We don't need to findandsinceincludes the factorsandvalues for which are given above.