hp40g+.book Page 66 Friday, December 9, 2005 1:03 AM

ILAP is the inverse Laplace transform of a given expression. Again, the expression is the value of a function of the variable stored in VX.

Laplace transform (LAP) and inverse Laplace transform (ILAP) are useful in solving linear differential equations with constant coefficients, for example:

y″ + p ⋅ y′ + q ⋅ y = f(x)

y(0) = a y′(0) = b

The following relations hold:

LAP(y)(x) = ∫+0 ∞ e–x⋅ ty(t)dt

ILAP(f)(x) =

1

⋅ ∫e

zx

f(z)dz

--------

 

 

2iπ

c

 

 

 

 

 

 

where c is a closed contour enclosing the poles of f.

The following property is used:

LAP(y′)(x) = – y(0) + x ⋅ LAP(y)(x)

The solution, y, of:

y″ + p ⋅ y′ + q ⋅ y = f(x), y(0) = a, y′(0) = b

is then:

⎛LAP(f(x)) + (x + p) ⋅ a + b⎞

ILAP⎝

------------------------------------------------------------------x2+ px + q

⎠

Example

To solve:

y″–6 ⋅ y′ + 9 ⋅ y = x ⋅ e3x, y(0) = a, y′(0) = b

c

type:

LAP(X · EXP(3 · X))

The result is:

1

--------------------------

x2 – 6x + 9

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Computer Algebra System (CAS)