Denoting by L the Laplace transform, you get the following:
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where C is a closed contour enclosing the poles of g.
The laplace command finds the Laplace transform of a function.
Examples.
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The ilaplace command finds the Laplace transform of a
function.
invlaplace is a synonym for ilaplace.
The Laplace transform has the following properties:
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These properties make the Laplace transform and inverse Laplace transform useful for solving linear differential equations with constant coefficients. For example, suppose you have
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then
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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:
Output:
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Input:
Output:
| ⎛ ⎝ | x3−18 x c0+6 x c1+6 c0 | ⎞ ⎠ | e3 x |
Note that this equation could be solved directly.
Input:
Output:
e3 x | ⎛ ⎝ | c0 x+c1 | ⎞ ⎠ | + |
| x3 e3 x |
You also can use the addtable command Laplacians of unspecified functions (see Section 6.26.2).