Laplace transform and inverse Laplace transform: laplace ilaplace invlaplace

6.57.2 Laplace transform and inverse Laplace transform: laplace ilaplace invlaplace

Denoting by L the Laplace transform, you get the following:

L(y)(x)

∫

+∞

e^−x uy(u)du

L⁻¹(g)(x)

2iπ

∫

e^z xg(z)dz

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:
1

x²+1
Input:
laplace(sin(t),t)
Output:
1

t²+1
Input:
laplace(sin(t),t,s)
Output:
1

s²+1

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 e^3 x, y(0)=c_0, y^′(0)=c_1

Here, p=−6, q=9.
Input:

laplace(x*exp(3*x))

Output:

x²−6 x+9

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

⎞
⎠

e^3 x

Note that this equation could be solved directly.
Input:

desolve(y’’-6*y’+9*y=x*exp(3*x),y)

Output:

e^3 x

⎛
⎝

c₀ x+c₁

⎞
⎠

x³ e^3 x

You also can use the addtable command Laplacians of unspecified functions (see Section 6.26.2).