Difference between revisions of "Laplace transform"
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The '''Laplace Transform''' of a function is a [[linear transformation]] from the space of <math>\Re \to \Re</math> (<math>\Re</math> is the space of integratable functions) defined as | The '''Laplace Transform''' of a function is a [[linear transformation]] from the space of <math>\Re \to \Re</math> (<math>\Re</math> is the space of integratable functions) defined as | ||
| − | <cmath>\ | + | <cmath>\pounds \{ f \} (s) = F(s) = \int _{0} ^ {\infty} e^{-st} f(t) dt</cmath> |
==Uses== | ==Uses== | ||
| Line 8: | Line 8: | ||
We first take the Laplace Transform of the equation, then solve the resulting algebra equation for <math>Y(s)</math>, and then take the inverse Laplace Transform of <math>Y(s)</math> to get <math>y</math>. | We first take the Laplace Transform of the equation, then solve the resulting algebra equation for <math>Y(s)</math>, and then take the inverse Laplace Transform of <math>Y(s)</math> to get <math>y</math>. | ||
| + | Here is a convenient table for reference: | ||
| + | |||
| + | {| class="wikitable" | ||
| + | |+ Table of Laplace Tranforms | ||
| + | |- <!-- Start of a new row --> | ||
| + | | style =”width:75%”| 1. <math>1</math> || <cmath>\frac{1}{s}, s>0</cmath> | ||
| + | |- | ||
| + | | 2. <math>e^{at}</math> || <cmath>\frac{1}{s-a}, s>a</cmath> | ||
| + | |- | ||
| + | | 3. <math>t^n, \text{n is a positive integer}</math> || <cmath>\frac{n!}{s^{n+1}},s>0</cmath> | ||
| + | |- | ||
| + | | 4. <math>t^p, p>-1</math> || <cmath>\frac{\Gamma (p+1)}{s^{p+1}}, s>0</cmath> | ||
| + | |- | ||
| + | | 5. <math>\sin (at)</math> || <cmath>\frac{a}{s^2+a^2},s>0</cmath> | ||
| + | |- | ||
| + | | 6. <math>\cos (at)</math> || <cmath>\frac{s}{s^2+a^2}, s>0</cmath> | ||
| + | |- | ||
| + | | 7. <math>\sinh (at)</math> || <cmath>\frac{a}{s^2-a^2}, s>|a|</cmath> | ||
| + | |- | ||
| + | | 8. <math>\cosh (at)</math> || <cmath>\frac{s}{s^2-a^2}, s>|a|</cmath> | ||
| + | |- | ||
| + | | 9. <math>\text{Heaviside} (t-a) = \begin{cases} 1 &\text{if } t \ge a \\ 0 &\text{if } t < a \end{cases}</math> || <cmath>\frac{e^{-as}}{s}, s>0</cmath> | ||
| + | |- | ||
| + | | 10. <math>\text{Heaviside} (t-a) \cdot f(t-a)</math> || <cmath>F(s-a)</cmath> | ||
| + | |- | ||
| + | |||
| + | |} | ||
| + | |||
| + | ==Example== | ||
| + | Solve the differential equation | ||
| + | |||
| + | <cmath>y''+y= H(t-5)</cmath> | ||
| + | |||
| + | , where <math>H(t)</math> is the Heaviside function, defined as | ||
| + | |||
| + | <cmath>H(x)= \begin{cases} x^2 &\text{if } x \ge 0 \\ x &\text{if } x < 0 \end{cases}</cmath> | ||
| + | |||
| + | ''Solution:'' | ||
| + | |||
| + | We take the Laplace transe | ||
==See Also== | ==See Also== | ||
*[[Differential equations]] | *[[Differential equations]] | ||
Latest revision as of 21:55, 12 January 2025
The Laplace Transform of a function is a linear transformation from the space of 
(
is the space of integratable functions) defined as
![\[\pounds \{ f \} (s) = F(s) = \int _{0} ^ {\infty} e^{-st} f(t) dt\]](http://latex.artofproblemsolving.com/c/0/9/c09a9374c66f75810eff10e76a71174c932b8031.png)
Uses
The Laplace Transform is a technique used to solve differential equation when some of the coefficients are not continuous functions.
We first take the Laplace Transform of the equation, then solve the resulting algebra equation for 
, and then take the inverse Laplace Transform of to get 
.
Here is a convenient table for reference:
1.  |
![\[\frac{1}{s}, s>0\]](http://latex.artofproblemsolving.com/7/6/6/766a4cbe25cde806229cc67dfa3cc138b3efbf59.png)
|
2.  |
![\[\frac{1}{s-a}, s>a\]](http://latex.artofproblemsolving.com/e/8/f/e8f07c58adc2a5beb5df9aba13f264bf6a6cdd2d.png)
|
3.  |
![\[\frac{n!}{s^{n+1}},s>0\]](http://latex.artofproblemsolving.com/f/9/f/f9fea45e965cc68020cc18b1c8d142219b867ee2.png)
|
4.  |
![\[\frac{\Gamma (p+1)}{s^{p+1}}, s>0\]](http://latex.artofproblemsolving.com/e/f/e/efef321db369d2a4cb39a89ef0721fb0507e418d.png)
|
5.  |
![\[\frac{a}{s^2+a^2},s>0\]](http://latex.artofproblemsolving.com/6/c/f/6cfdf63edb2f560ada85d08d6c5b966c37914a43.png)
|
6.  |
![\[\frac{s}{s^2+a^2}, s>0\]](http://latex.artofproblemsolving.com/5/7/0/570d81d447348238398bbc3e033aa1cddd12a695.png)
|
7.  |
![\[\frac{a}{s^2-a^2}, s>|a|\]](http://latex.artofproblemsolving.com/b/e/b/beba120e52d3a38ad1ba3e3e9a207e593831188f.png)
|
8.  |
![\[\frac{s}{s^2-a^2}, s>|a|\]](http://latex.artofproblemsolving.com/3/c/1/3c166ce1c68d54f8990b12ec6c8d24e2c2dab47b.png)
|
9.  |
![\[\frac{e^{-as}}{s}, s>0\]](http://latex.artofproblemsolving.com/e/b/7/eb75325891779d5e27711d5a656c63215d6e9850.png)
|
10.  |
![\[F(s-a)\]](http://latex.artofproblemsolving.com/2/9/9/299858820468af76959c7e8ade3ba1d3c177294d.png)
|
Example
Solve the differential equation
![\[y''+y= H(t-5)\]](http://latex.artofproblemsolving.com/4/c/e/4cedbf0f76a7b1b1adae5edb9d290f2bf4ddd521.png)
, where 
is the Heaviside function, defined as
![\[H(x)= \begin{cases} x^2 &\text{if } x \ge 0 \\ x &\text{if } x < 0 \end{cases}\]](http://latex.artofproblemsolving.com/e/b/e/ebe60b118092d96e24e35ab702131b3aae4f4f6f.png)
Solution:
We take the Laplace transe
See Also
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