Nonhomogeneous PDE with non-constant coefficients

In summary, the conversation discusses finding the general solution to a nonhomogeneous PDE with non-constant coefficients and an "u" term. The method of change of variables is suggested, specifically using {x = X, y = YX} to transform the PDE into an ODE with solution u(X,Y) = -1+1/Y+XF(Y). The motivation behind this particular change of variables is not explained.
  • #1
kingwinner
1,270
0
This is a question from a book in which I can't figure out, but it has no solutions at the back.

Find the general solution to the PDE:
xy ux + y2 (uy) - y u = y - x

I've learned methods such as change of variables and characteristic curves, but I'm not sure how I can apply them in this situation. The PDE is nonhomogeneous, with non-constant coefficients, and there is an "u" term.

Can somebody help?

Thank you!
 
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  • #2
If change variables as {x = X, y = YX} we get ODE

XYu_X-Yu-Y+1=0.

Its solution is

u(X,Y) = -1+1/Y+XF(Y),

where F(Y) is an arbitrary function.
So the general solution to initial PDE is as follows

u(x,y) = -1+x/y+xF(y/x)
 
  • #3
Thanks, but what motivates that particular change of variables? (i.e. how did you derive them?) How did you get x = X and y = YX? (Is there a systematic way to derive these?)
 

1. What is a nonhomogeneous PDE with non-constant coefficients?

A nonhomogeneous PDE (partial differential equation) with non-constant coefficients is a type of mathematical equation that involves multiple variables and their partial derivatives. It is called "nonhomogeneous" because the equation includes terms that are not equal to zero, and the coefficients (numbers in front of the variables) are not constant, meaning they can change depending on the values of the variables.

2. How is a nonhomogeneous PDE with non-constant coefficients different from a homogeneous PDE?

A homogeneous PDE has all terms equal to zero and constant coefficients, while a nonhomogeneous PDE has at least one term that is not equal to zero and non-constant coefficients. This makes the solution to a nonhomogeneous PDE more complex, as it involves finding a particular solution in addition to the general solution.

3. What are some examples of real-life applications of nonhomogeneous PDEs with non-constant coefficients?

Nonhomogeneous PDEs with non-constant coefficients are used in various fields of science and engineering, such as fluid mechanics, heat transfer, electromagnetism, and quantum mechanics. For example, they can be used to model the flow of air or water in a pipe, the distribution of temperature in a heated object, the behavior of electromagnetic waves, and the movement of particles on a quantum level.

4. How do you solve a nonhomogeneous PDE with non-constant coefficients?

The general method for solving a nonhomogeneous PDE with non-constant coefficients is to first find the general solution by setting all coefficients to zero. Then, a particular solution must be found by using boundary conditions or initial values. This can be done analytically (by hand) or numerically (using a computer). Finally, the general and particular solutions are combined to get the final solution to the PDE.

5. What are some techniques for solving nonhomogeneous PDEs with non-constant coefficients?

Some common techniques for solving nonhomogeneous PDEs with non-constant coefficients include separation of variables, Fourier transform, Laplace transform, and method of characteristics. Each technique has its own advantages and is used depending on the specific form of the PDE and the boundary/initial conditions given. In some cases, a combination of these techniques may be needed to find the solution.

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