Difference between revisions of "Laplace Transform to ODEs"

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(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...")
 
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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]].
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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:
 
First consider the following property of the Laplace transform:
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:<math>\mathcal{L}\{f''\}=s^2\mathcal{L}\{f\}-sf(0)-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
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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>
 
:<math>\mathcal{L}\{f^{(n)}\}=s^n\mathcal{L}\{f\}-\sum_{i=1}^{n}s^{n-i}f^{(i-1)}(0)</math>
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:<math>f^{(i)}(0)=c_i</math>
 
:<math>f^{(i)}(0)=c_i</math>
  
Using the [[linearity]] of the Laplace transform it is equivalent to rewrite the equation as
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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>
 
:<math>\sum_{i=0}^{n}a_i\mathcal{L}\{f^{(i)}(t)\}=\mathcal{L}\{\phi(t)\}</math>
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:<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>
 
:<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>
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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.
 
Note that if the initial conditions are all zero, i.e.

Revision as of 10:21, 29 October 2021

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:

One can prove by induction that

Now we consider the following differential equation:

with given initial conditions

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

obtaining

Solving the equation for and substituting with one obtains

The solution for f(t) is obtained by applying the inverse Laplace transform to

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

then the formula simplifies to

An example

We want to solve


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

We note that

and we get

The equation is then equivalent to

We deduce

Now we apply the Laplace inverse transform to get

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