Transforming Partial Differential Equations into Constant Coefficient Form

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SUMMARY

The discussion focuses on transforming partial differential equations (PDEs) into constant coefficient form using variable transformations. The user, Chet, attempts to apply transformations such as x1=(U0*beta + x*Ud) and d/dx=((U0*beta/x)+Ud) d/dx1 to achieve this goal. The need for a suitable equation for the variable transformation is emphasized, along with the importance of adjusting boundary conditions and finding solutions. The inquiry also raises the question of whether phi is a function of x and y, indicating a deeper exploration of the relationship between variables.

PREREQUISITES
  • Understanding of partial differential equations (PDEs)
  • Familiarity with variable transformations in mathematical analysis
  • Knowledge of boundary conditions in differential equations
  • Experience with mathematical functions and their dependencies
NEXT STEPS
  • Research methods for transforming PDEs into constant coefficient form
  • Explore the implications of boundary conditions on variable transformations
  • Study the role of functions in PDEs, specifically the relationship between phi, x, and y
  • Investigate advanced techniques in mathematical analysis for solving PDEs
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Mathematicians, physics students, and engineers dealing with partial differential equations, particularly those interested in variable transformations and boundary value problems.

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Homework Statement


The problem statement can be expressed in one of these forms listed in order of preference.
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upload_2015-1-7_3-59-23.jpeg
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Every character with exception of x, y, t, and C are constants.

Homework Equations


I require a change of variable or series of subsequent change of variables that can convert anyone of these equations into an equation having constant coefficients.

The Attempt at a Solution


I have tried x1=(U0*beta + x*Ud)
I have also tried d/dx=((U0*beta/x)+Ud) d/dx1 and d/dy=y d/dy1

where x1 and y1 are the new variables for x and y respectively.
I know how to do the rest, like changing the boundary conditions and finding an a solution, I just need an appropriate equation for my variable transformation.
Any comment at all (including things like this equation is not solvable and why that is so) regarding the problem will be welcomed.
 
Is phi a function of x and y?

Chet
 

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