5.57.2 Laplace transform and inverse Laplace transform: laplace ilaplace invlaplace
Denoting by L the Laplace transform,
you get the following:
where C is a closed contour enclosing the poles of g.
The laplace command finds the Laplace transform of a function.
-
laplace takes one mandatory argument and two optional
arguments:
-
expr, an expression involving a variable.
- Optionally, x, the variable name (by default x).
- Optionally, s, a variable for the output (by default x).
- laplace(expr ⟨ x, ⟩)
returns the Laplace transform of expr.
Examples
-
Input:
laplace(sin(x))
Output:
- Input:
laplace(sin(t),t)
Output:
- Input:
laplace(sin(t),t,s)
Output:
The ilaplace command finds the Laplace transform of a
function.
invlaplace is a synonym for ilaplace.
-
ilaplace takes one mandatory argument and two optional
arguments:
-
expr, an expression involving a variable.
- Optionally, x, the variable name (by default x).
- Optionally, s, a variable for the output (by default x).
- ilaplace(expr ⟨ x, ⟩)
returns the inverse Laplace transform of expr.
The Laplace transform has the following properties:
L(y′)(x) | = | −y(0)+xL(y)(x) |
L(y″)(x) | = | −y′(0)+xL(y′)(x) |
| = | −y′(0)−xy(0)+x2L(y)(x)
|
|
These properties make the Laplace transform and inverse Laplace
transform useful for solving linear differential equations
with constant coefficients. For example, suppose you have
| | y′′ +p y′ +q y = f(x) | | | | | | | | | |
| y(0)=a, y′(0)=b
| | | | | | | | | |
|
then
L(f)(x) | = | L(y″+py′+qy)(x) |
| = | −y′(0)−x y(0)+x2 L(y)(x)−p y(0)+p x L(y)(x))+q L(y)(x) |
| = | (x2+p x+q) L(y)(x)−y′(0)−(x+p) y(0)
|
|
Therefore, if a=y(0) and b=y′(0), you get
L(f)(x)=(x2+p x+q)L(y)(x)−(x+p) a−b
|
and the solution of the differential equation is:
y(x)= L−1((L(f)(x)+(x+p) a +b)/(x2+p x+q))
|
Example
Solve:
y′′ −6 y′+9 y = x e3 x,
y(0)=c_0, y′(0)=c_1
|
Here, p=−6, q=9.
Input:
laplace(x*exp(3*x))
Output:
Input:
ilaplace((1/(x^2-6*x+9)+(x-6)*c_0+c_1)/(x^2-6*x+9))
Output:
| | ⎛
⎝ | x3−18 x c0+6 x c1+6 c0 | ⎞
⎠ | e3 x
|
Note that this equation could be solved directly.
Input:
desolve(y’’-6*y’+9*y=x*exp(3*x),y)
Output:
e3 x | ⎛
⎝ | c0 x+c1 | ⎞
⎠ | + | | x3 e3 x
|
You also can use the addtable command
Laplacians of unspecified functions (see Section 5.26.2).