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
whenand
when
AssumingthenandSubstitute these into the original equation to get
We divide by the none zero factorto obtain
which factorises to give
We obtain the solution
(twice).
The solution is then
We can findandusing the conditions given.
so the second condition
The solution is then
Example: Solve the equation
subject to the conditionswhenandwhen
AssumingthenandSubstitute these into the original equation to get
We divide by the none zero factorto obtain
which has the solutions
where
The solution is then
We can findandusing the conditions given.
We don't need to findandsinceincludes the factors
and
values for which are given above.
