- #1
jomomma
- 4
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http://mathbin.net/906
cant figure this one out
cant figure this one out
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Basic PDE Help: Simplifying the Confusing Concepts | MathBin
- Context: Undergrad
- Thread starter jomomma
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Discussion Overview
The discussion revolves around solving a partial differential equation (PDE) of the form \( x\frac{\partial u}{\partial x}+2y\frac{\partial u}{\partial y} + \frac{\partial u}{\partial z} = 3u \). Participants are exploring methods for simplifying and solving this equation, particularly through the method of characteristics.
Discussion Character
- Technical explanation
- Mathematical reasoning
- Debate/contested
Main Points Raised
- One participant expresses confusion about how to approach the PDE and seeks assistance.
- Another participant questions the initial approach taken and asks for clarification on what has been attempted, specifically regarding the characteristics method.
- A participant shares their work on the characteristics equations, detailing the relationships derived for \( x, y, z, \) and \( u \) over time.
- There is a proposal for a potential solution in the form \( u(x,y,z)=\varphi(xe^{-z},ye^{-2z})e^{3z} \), but it is presented as a question for verification.
- Another participant suggests verifying the proposed solution by substituting it back into the original PDE.
- A participant acknowledges they can check the solution but seeks confirmation from others to ensure completeness and correctness.
Areas of Agreement / Disagreement
Participants do not reach a consensus on the correctness of the proposed solution, and there is ongoing uncertainty regarding the completeness of the approach and the potential for errors in the calculations.
Contextual Notes
The discussion includes various assumptions about the characteristics method and the handling of the \( u_z \) term, which remain unresolved. There is also a lack of clarity on whether all possible solutions have been considered.
- #2
tiny-tim
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Welcome to PF!
Hi jomomma! Welcome to PF!
(type "tex" instead of "EQ" on this forum
)
What have you tried?
jomomma said:
Hi jomomma! Welcome to PF!
(type "tex" instead of "EQ" on this forum
)What have you tried?
- #3
HallsofIvy
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What do you mean by "figure this out"? Did you stare at it expecting the answer to suddenly pop into your head or did you DO something like writing down the equations of the characteristics?
If the latter, please show us what you did!
If the latter, please show us what you did!
- #4
jomomma
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I tried doing characteristics but i am getting confused with the u_z term. Heres what i do
<br /> <br /> \[<br /> \frac{dx}{dt}&=&x\Rightarrow x=x_0e^t<br /> \]<br /> \[<br /> \frac{dy}{dt}&=&2y\Rightarrow y=y_0e^2t<br /> \]<br /> \[<br /> \frac{dz}{dt}&=&1\Rightarrow z=z_0 + t<br /> \]<br /> \[<br /> \frac{du}{dt}&=&3u\Rightarrow u=u_0e^3t<br /> \]<br /> <br />
from there i try to put y and x into the equation for u
<br /> u=xy=x_0y_0e^3t<br />
what do i do with z and \varphi(x,y)
<br /> <br /> \[<br /> \frac{dx}{dt}&=&x\Rightarrow x=x_0e^t<br /> \]<br /> \[<br /> \frac{dy}{dt}&=&2y\Rightarrow y=y_0e^2t<br /> \]<br /> \[<br /> \frac{dz}{dt}&=&1\Rightarrow z=z_0 + t<br /> \]<br /> \[<br /> \frac{du}{dt}&=&3u\Rightarrow u=u_0e^3t<br /> \]<br /> <br />
from there i try to put y and x into the equation for u
<br /> u=xy=x_0y_0e^3t<br />
what do i do with z and \varphi(x,y)
Last edited:
- #5
jomomma
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could someone verify whether the correct answer is
<br /> <br /> u(x,y,z)=\varphi(xe^{-z},ye^{-2z})e^{3z}<br /> <br />
<br /> <br /> u(x,y,z)=\varphi(xe^{-z},ye^{-2z})e^{3z}<br /> <br />
- #6
NoMoreExams
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Can't you plug it in and check like you would with any DE?
- #7
jomomma
- 4
- 0
yes, i can and i have, but i wanted third party confirmation, because i may not have included all solutions or may have made a mistake
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