[This example would have been presented at the tutorial
of 2008 March 13 (Thursday)
that was cancelled due to a winter storm.]
Find the complete solution to the initial value problem
The right side function
8 e-3x
is of a form for which the method of undetermined
coefficients may be used to find the particular solution.
However, e-3x
is part of the complementary function.
Substitute this trial particular solution into the original
ODE:
The general solution is therefore
Imposing the initial conditions:
One quickly finds that the only solution is
A = B = 0
The complete solution is therefore
Additional Notes [not covered in the tutorial]:
One may use the method of variation of parameters
to find the particular solution:
The general solution of the linear ODE is the sum of the
complementary function and the particular solution:
Alternative Solution Using Laplace Transforms:
Let
then the initial value problem transforms to
In this case, the method of Laplace transforms is by far the
fastest method
(provided one spots the cancellation that
avoids the need for partial fractions).