Laplace Transform to ODEs - Revision history

Khanh at 18:25, 27 November 2021

2021-11-27T18:25:25Z

Lila at 16:21, 29 October 2021

2021-10-29T16:21:34Z

Lila: Created page with "In mathematics, the Laplace transform is a powerful integral transform used to switch a function from the time domain to the Laplace transform#s-domain equiv..."

2021-10-29T16:20:30Z

Created page with "In mathematics, the Laplace transform is a powerful integral transform used to switch a function from the time domain to the Laplace transform#s-domain equiv..."

New page

In [[mathematics]], the [[Laplace transform]] is a powerful [[integral transform]] used to switch a function from the [[time domain]] to the [[Laplace transform#s-domain equivalent circuits and impedances|s-domain]]. The Laplace transform can be used in some cases to solve [[linear differential equation]]s with given [[Initial value problem|initial conditions]].

First consider the following property of the Laplace transform:

:<math>\mathcal{L}\{f'\}=s\mathcal{L}\{f\}-f(0)</math>

:<math>\mathcal{L}\{f''\}=s^2\mathcal{L}\{f\}-sf(0)-f'(0)</math>

One can prove by [[Mathematical induction|induction]] that

:<math>\mathcal{L}\{f^{(n)}\}=s^n\mathcal{L}\{f\}-\sum_{i=1}^{n}s^{n-i}f^{(i-1)}(0)</math>

Now we consider the following differential equation:

:<math>\sum_{i=0}^{n}a_if^{(i)}(t)=\phi(t)</math>

with given initial conditions

:<math>f^{(i)}(0)=c_i</math>

Using the [[linearity]] of the Laplace transform it is equivalent to rewrite the equation as

:<math>\sum_{i=0}^{n}a_i\mathcal{L}\{f^{(i)}(t)\}=\mathcal{L}\{\phi(t)\}</math>

obtaining

:<math>\mathcal{L}\{f(t)\}\sum_{i=0}^{n}a_is^i-\sum_{i=1}^{n}\sum_{j=1}^{i}a_is^{i-j}f^{(j-1)}(0)=\mathcal{L}\{\phi(t)\}</math>

Solving the equation for <math> \mathcal{L}\{f(t)\}</math> and substituting <math>f^{(i)}(0)</math> with <math>c_i</math> one obtains

:<math>\mathcal{L}\{f(t)\}=\frac{\mathcal{L}\{\phi(t)\}+\sum_{i=1}^{n}\sum_{j=1}^{i}a_is^{i-j}c_{j-1}}{\sum_{i=0}^{n}a_is^i}</math>

The solution for ''f''(''t'') is obtained by applying the [[inverse Laplace transform]] to <math>\mathcal{L}\{f(t)\}.</math>

Note that if the initial conditions are all zero, i.e.

:<math>f^{(i)}(0)=c_i=0\quad\forall i\in\{0,1,2,...\ n\}</math>

then the formula simplifies to

:<math>f(t)=\mathcal{L}^{-1}\left\{{\mathcal{L}\{\phi(t)\}\over\sum_{i=0}^{n}a_is^i}\right\}</math>

==An example==
We want to solve

:<math>f''(t)+4f(t)=\sin(2t)</math>

with initial conditions ''f''(0) = 0 and ''f′''(0)=0.

We note that

:<math>\phi(t)=\sin(2t)</math>

and we get

:<math>\mathcal{L}\{\phi(t)\}=\frac{2}{s^2+4}</math>

The equation is then equivalent to

:<math>s^2\mathcal{L}\{f(t)\}-sf(0)-f'(0)+4\mathcal{L}\{f(t)\}=\mathcal{L}\{\phi(t)\}</math>

We deduce

:<math>\mathcal{L}\{f(t)\}=\frac{2}{(s^2+4)^2}</math>

Now we apply the Laplace inverse transform to get

:<math>f(t)=\frac{1}{8}\sin(2t)-\frac{t}{4}\cos(2t)</math>

==Licensing==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Laplace_transform_applied_to_differential_equations Laplace transform applied to ODEs, Wikipedia] under a CC BY-SA license

@@ Line 68: / Line 68: @@
 :<math>f(t)=\frac{1}{8}\sin(2t)-\frac{t}{4}\cos(2t)</math>
 ==Licensing==
 Content obtained and/or adapted from:
 * [https://en.wikipedia.org/wiki/Laplace_transform_applied_to_differential_equations Laplace transform applied to ODEs, Wikipedia] under a CC BY-SA license

@@ Line 1: / Line 1: @@
-In [[mathematics]], the [[Laplace transform]] is a powerful [[integral transform]] used to switch a function from the [[time domain]] to the [[Laplace transform#s-domain equivalent circuits and impedances|s-domain]]. The Laplace transform can be used in some cases to solve [[linear differential equation]]s with given [[Initial value problem|initial conditions]].
+The Laplace transform is a powerful integral transform used to switch a function from the time domain to the s-domain. The Laplace transform can be used in some cases to solve linear differential equations with given initial conditions.
 First consider the following property of the Laplace transform:
@@ Line 7: / Line 7: @@
 :<math>\mathcal{L}\{f''\}=s^2\mathcal{L}\{f\}-sf(0)-f'(0)</math>
-One can prove by [[Mathematical induction|induction]] that
+One can prove by induction that
 :<math>\mathcal{L}\{f^{(n)}\}=s^n\mathcal{L}\{f\}-\sum_{i=1}^{n}s^{n-i}f^{(i-1)}(0)</math>
@@ Line 19: / Line 19: @@
 :<math>f^{(i)}(0)=c_i</math>
-Using the [[linearity]] of the Laplace transform it is equivalent to rewrite the equation as
+Using the linearity of the Laplace transform it is equivalent to rewrite the equation as
 :<math>\sum_{i=0}^{n}a_i\mathcal{L}\{f^{(i)}(t)\}=\mathcal{L}\{\phi(t)\}</math>
@@ Line 31: / Line 31: @@
 :<math>\mathcal{L}\{f(t)\}=\frac{\mathcal{L}\{\phi(t)\}+\sum_{i=1}^{n}\sum_{j=1}^{i}a_is^{i-j}c_{j-1}}{\sum_{i=0}^{n}a_is^i}</math>
-The solution for ''f''(''t'') is obtained by applying the [[inverse Laplace transform]] to <math>\mathcal{L}\{f(t)\}.</math>
+The solution for ''f''(''t'') is obtained by applying the inverse Laplace transform to <math>\mathcal{L}\{f(t)\}.</math>
 Note that if the initial conditions are all zero, i.e.