Laplace transform for solving ODE with variable coefficients

In summary, the conversation discusses the theoretical use of Laplace transform to solve ODEs with variable coefficients. The process involves finding the Laplace transform of f(x)y''(x) and using integration by parts to reduce it to an algebraic equation. However, this method can be challenging for general variable coefficients. There is a typo in the conversation and the last part may need further clarification.
  • #1
angelas
8
0
Can we use laplace transform to solve an ODE with variable coefficients?

Like this one:

4x y" + 2 y' + y = exp (-x)
 
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  • #2
Theoretically, yes, if you can find the Laplace transform of f(x)y"(x)!

The Laplace transform of xy"(x) is, by definition,
[tex]\int_0^\infty e^{-st}ty"(t)dt[/tex]
Integrate by parts with [itex]u= e^{-st}t[/itex] so that [itex]du= (e^{-st}- st e^{-st})dt[/itex], dv= y"(t)dt so that v= y'(t). Then
[tex]-\int_0^\infty (e^{-st}- ste^{-st})y'(t)dt[/tex]
Do that by integration by parts with [itex]u= (e^{-st}- ste^{-st}[/itex] so that [itex]du= -2e^{-st}+ s^2te^{-st}[/itex], dv= y'(t)dt so that v= y. Then
[tex]2y(0)+ \int_0^\infty (2e^{-st}- s^2te^{-st})y(t)dt[/itex]
Write that last as a Laplace transform of y and reduce to an algebraic equation as usual.

Unfortunately, that is not always easy to do with general variable coefficients. The Laplace transform is basically a method for very mechanically solving linear equations with constant coefficients.
 
  • #3
HallsofIvy said:
Theoretically, yes, if you can find the Laplace transform of f(x)y"(x)!

The Laplace transform of xy"(x) is, by definition,
[tex]\int_0^\infty e^{-st}ty"(t)dt[/tex]
Integrate by parts with [itex]u= e^{-st}t[/itex] so that [itex]du= (e^{-st}- st e^{-st})dt[/itex], dv= y"(t)dt so that v= y'(t). Then
[tex]-\int_0^\infty (e^{-st}- ste^{-st})y'(t)dt[/tex]
Do that by integration by parts with [itex]u= (e^{-st}- ste^{-st}[/itex] so that [itex]du= -2e^{-st}+ s^2te^{-st}[/itex], dv= y'(t)dt so that v= y. Then
[tex]2y(0)+ \int_0^\infty (2e^{-st}- s^2te^{-st})y(t)dt[/itex]
Write that last as a Laplace transform of y and reduce to an algebraic equation as usual.
Thanks so much.
I think there is a typo here: du = -2e^{-st}+ s^2te^{-st}

it should be du = -2 s e^{-st}+ s^2te^{-st}

and also about the last part why we get 2 y(0). I think it should be zero.

and also how can I write s^2 t e ^(-st) y as the laplace transform of y? is it equal to the laplace transform of s^2ty?

Sorry if I ask too many questions.
 
Last edited:

1. What is the Laplace transform?

The Laplace transform is a mathematical tool used to solve differential equations by transforming them from the time domain to the frequency domain. It is represented by the integral of a function multiplied by an exponential term, and it allows for the simplification of complex differential equations.

2. How can the Laplace transform be used to solve ODEs with variable coefficients?

The Laplace transform can be applied to differential equations with variable coefficients by first transforming the equation into the frequency domain. This results in a simpler algebraic equation, which can then be solved using standard techniques. Once the solution is found, it can be transformed back into the time domain to obtain the solution to the original ODE.

3. What types of differential equations can be solved using the Laplace transform?

The Laplace transform can be used to solve a wide range of linear differential equations, including ordinary differential equations (ODEs) and partial differential equations (PDEs). It is particularly useful for solving ODEs with constant or variable coefficients, as well as initial value problems.

4. What are the advantages of using the Laplace transform for solving ODEs with variable coefficients?

One of the main advantages of using the Laplace transform for solving ODEs with variable coefficients is that it allows for the simplification of complex equations and reduces the problem to a simpler algebraic form. Additionally, it can be used to solve a wide range of differential equations, making it a versatile tool for scientists and engineers.

5. Are there any limitations to using the Laplace transform for solving ODEs with variable coefficients?

Although the Laplace transform is a powerful tool for solving ODEs with variable coefficients, it does have some limitations. It can only be applied to linear equations, and it may not always be possible to find an inverse transform to obtain the solution in the time domain. Additionally, the transformation process can be time-consuming for more complex equations.

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