The Laplace Trans/ orm Method 247Using partial fraction expansion, results in(lO) £[ (x)] __ l __ ~_l __ ~_s __^17 1 1 (s - 2)
Y - s - 1 8 s^2 + 1 8 s^2 + 1 8 ( s - 2)^2 + 1+8 ( s - 2)2 + 1 ·Apply the Inverse Laplace Transform
Applying the inverse Laplace transform to (10) and using linearity of the
inverse Laplace transform, yieldsyx ( ) =£ -1 [ --^1 ] --£^1 -1 [ --^1 ] --£^1 -1 [ --s ]
s - 1 8 s^2 + 1 8 s^2 + 1= e x - -^1 sm. x - -^1 cos x - -1 7 e 2x sm. x + -^1 e 2x cos x
8 8 8 8.Hence, the solution of the given initial value problem isy ( ) x = e x - -^1 sm. x - -^1 cos x - 1 7 - e^2 x sm. x + - e^1 2x cos x.
8 8 8 8!Comments on Computer Software! Manually using the Laplace trans-
form method to solve a linear differential equation with constant coefficients
or an initial value problem which includes such a differential equation is a
three step process. First, one calculates the Laplace transform of the differ-
ential equation and substitutes for the initial conditions, if any are specified.
Next, one uses partial fraction expansion, a table of Laplace transforms and
their inverses, and some algebraic rearrangement, when necessary, so that the
linearity property of the Laplace transform can be used. Finally, the inverse
Laplace transform is determined. Some computer algebra systems (CAS) in-
clude one or more algorithms for using the Laplace transform method to solve
linear differential equations and initial value problems. In one method of solu-
tion, the user follows the three steps listed above. That is , the user instructs
the CAS to calculate the Laplace transform of the given differential equation
and specifies the initial conditions, if specific values are given. Next, the user
has the CAS perform a partial fraction expansion. And then, the user has the
CAS calculate the inverse Laplace transform to obtain the required solution.
In the second method, the user enters the differential equation and initial
conditions, if any, and instructs the CAS to solve the differential equation
or initial value problem using the Laplace transform method. In this case,
the CAS will not show the step-by-step computations used to arrive at the
answer, it will simply provide the answer, if it can. Thus, if a user specifies