## 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 form The solution will contain two arbitrary constants, which can be found given two boundary conditions on or some derivative of Example: Solve the equation subject to the conditions when and when Assuming then and Substitute these into the original equation to get We divide by the none zero factor to obtain which factorises to give We obtain the two solutions The solution is then We can find and using the conditions given. (1) (2)

(1)+(2) gives then from (1) so Example: Solve the equation subject to the conditions when and when Assuming then and Substitute these into the original equation to get We divide by the none zero factor to obtain which factorises to give We obtain the solution (twice).

The solution is then We can find and using the conditions given.  so the second condition  The solution is then Example: Solve the equation subject to the conditions when and when Assuming then and Substitute these into the original equation to get We divide by the none zero factor to obtain which has the solutions where The solution is then  We can find and using the conditions given.  We don't need to find and since includes the factors and values for which are given above.  