Is every one playing a joke on me? PDE solution question

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Discussion Overview

The discussion revolves around the solutions to partial differential equations (PDEs), specifically focusing on the emergence of sine and cosine functions in the context of separation of variables. Participants explore the relationship between PDEs and ordinary differential equations (ODEs), as well as the underlying principles that lead to these solutions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Homework-related

Main Points Raised

  • One participant expresses confusion about the origin of sine and cosine functions in the solutions of PDEs after separation of variables.
  • Another participant points out that separating variables leads to several ODEs and questions whether one of them resembles a harmonic oscillator equation.
  • A participant identifies the heat equation and presents its transformation into ODEs, noting the specific forms of the solutions involving sine, cosine, and exponential functions.
  • One reply suggests that the author of the material assumes these solutions are well-known and provides a method for deriving them through integration.
  • Several participants emphasize the necessity of understanding ODEs before tackling PDEs, indicating that the solutions presented are common forms encountered in ODE studies.
  • There is a mention of potential confusion regarding hyperbolic sine and cosine functions in relation to the expected solutions.
  • A participant acknowledges their prior reading of ODEs but admits to possibly missing key concepts.

Areas of Agreement / Disagreement

Participants generally agree on the importance of understanding ODEs to grasp PDE solutions, but there is no consensus on the specific explanations for the emergence of sine and cosine functions or the completeness of the material on PDEs.

Contextual Notes

Some participants note that the discussion may lack clarity on the assumptions made in the derivation of solutions and the definitions of terms used, which could affect understanding.

knockout_artist
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Ok I have read/browse 3 books so far about PDEs.

They separate variables. Then they suddenly go.

solution is

X=
some thing sin
some thing cos

Y=
some thing -ek
some thing ek


Where do sine and cos comes from ?
How do they know?? where is it explained?
 
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After they separate variables,
you get several ODEs.
What are these ODEs?
Do you recognize the ODEs?
Is one of them a harmonic oscillator equation?
(Do you know the Euler identity, relating sin and cos to exp?)
 
The one I mentioned above was heat equation.

V2T = 0 (heat equation)

become this
X" = -k2X (k power 2)
Y" = k2y (k power 2)

Then it say solution of this equation is X= sin kx
cos kx

Y= e ky (e power ky)
e -ky (e power -ky)

And yes I do know Euler's

ei@ =cos@ + i sin@ ( @ = theta)
 
As robphy indicates, it's likely the author simply regards these as well known solutions.
One way to obtain the result is to multiply both sides of the equation by 2x':
2 x''.x' = -2k x.x'
Integrating:
x'^2 = -kx^2 + c
x' = sqrt(c-kx^2)
The RHS immediately suggests the substitution x = (sqrt(c/k)) sin @
 
You need to study ODEs before you study PDEs. These are simple examples of linear ODEs with constant coefficients. A book on PDEs shouldn't need to explain how to solve them.
 
As indicated above, you have to understand how to solve ODEs to understand PDEs. Those two solutions are really the most common form that you should have seen when studying ODEs.

The only reason I'm adding anything to this is that at first I thought your question was going to be how all of a sudden they threw a hyperbolic sine or cosine at you when you expected a solution of the form you already listed. That's when you'll have to realize that there are many solutions to your ODEs and picking the correct ones can simplify soliving your PDEs significantly.

Just a heads up so that you don't immediately come running back when they show you different solutions to the same ODEs that you'd been using.
 
Thank you S-happens, haruspex, robphy.

I look into it.

I did read ODEs a little, I think I must have missed stuff.
 
test.
 

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