Laplace DE separation of variables

In summary, the conversation discusses finding a nontrivial solution to a differential equation and the use of separation of variables. The correct solution is X(s) = C/(s+1)^4 and the individual is struggling with integrating using natural logs. The solution is to raise both sides to the power of e to remove the natural logs.
  • #1
accountkiller
121
0

Homework Statement


I'm supposed to find a nontrivial solution to tx'' + (t-2)x' + x = 0, x(0) = 0. You don't really need to know that but just in case.

I got to this point:
(s+1)X'(s) + 4X(s) = 0

Now I need to separate variables to find a solution but I've been working on this for two days and can't figure out how to get to the correct solution, which is:
X(s) = C/(s+1)^4

So my question is, how do I go from the first equation to the second using separation of variables?

Homework Equations





The Attempt at a Solution


I've been trying for two days now.. with X(s) = X and X'(s) = dX/ds, I separated all X's and dX's and the same with s and ds.. then when I took the integral of both sides, I get natural logs since I'm doing the integral of 1/x and 1/(s+1) ... I don't know where to go with natural logs, since that's not the answer.

Any help would be appreciated, thanks!
 
Physics news on Phys.org
  • #2
as you have natural log on both sides, you should be able to remove it by raising both sides to the power of e...
 
  • #3
Ah, right! That solves my entire dilemma! *phew* thanks so much for pointing that out ;)
 
  • #4
scratch that
 

What is Laplace DE separation of variables?

Laplace DE separation of variables is a method for solving partial differential equations (PDEs) that involves separating the variables in the equation and solving each part separately. This method is named after the French mathematician Pierre-Simon Laplace.

What types of PDEs can be solved using Laplace DE separation of variables?

Laplace DE separation of variables can be used to solve linear PDEs with constant coefficients, also known as homogeneous PDEs. This method is not applicable to nonlinear PDEs or PDEs with variable coefficients.

What is the process for solving a PDE using Laplace DE separation of variables?

The first step is to separate the variables in the PDE, usually into two or more parts. Then, each part is solved separately using techniques from ordinary differential equations. The solutions for each part are then combined to form the general solution to the PDE.

What are the benefits of using Laplace DE separation of variables?

This method is useful for solving PDEs that cannot be solved using other techniques, such as the method of characteristics or the method of characteristics. It also allows for the general solution to be found, rather than just a specific solution for given initial or boundary conditions.

Are there any limitations to Laplace DE separation of variables?

Yes, this method is only applicable to certain types of PDEs and cannot be used for nonlinear or variable coefficient PDEs. It also relies on the ability to separate the variables in the equation, which is not always possible.

Similar threads

  • Calculus and Beyond Homework Help
Replies
14
Views
240
  • Calculus and Beyond Homework Help
Replies
9
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
859
  • Calculus and Beyond Homework Help
Replies
1
Views
797
  • Calculus and Beyond Homework Help
Replies
1
Views
173
  • Calculus and Beyond Homework Help
Replies
1
Views
633
  • Calculus and Beyond Homework Help
Replies
3
Views
222
  • Calculus and Beyond Homework Help
Replies
8
Views
709
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
858
Back
Top