How to Solve the PDE 2Uxxy + 3Uxyy - Uxy = 0?

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In summary, the conversation discusses the substitution of W for Uxy and the use of a change of coordinates to solve Uxy=f(3x-2y)exp((2x+3y)/3). The expert provides a general solution for the PDE, which involves arbitrary functions and double integration. They also mention that there are various ways to solve the problem.
  • #1
mglaros
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2Uxxy+3Uxyy-Uxy=0 where U=U(x,y)

I made the substitution W=Uxy and then used a change of coordinates (n= 2x+3y, and r=3x-2y) which reduced the problem to solving Uxy=f(3x-2y)exp((2x+3y)/3) because W=f(r)exp(n/3). Now I have no idea where to go from there. Any help would be much appreciated.

Thanks
 
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  • #2
The general solution to your PDE is as follows

U(x,y) = F1(x)+F2(y)+exp(x/2)F3(3x-2y),

where F1, F2, F3 are arbitrary functions.
 
  • #3
kosovtsov,

Would you mind explaining to me how you arrived at that solution? It would be greatly appreciated.

Thanks
 
  • #4
There is a general principle. If a problem can be solved, it as a rule, can be solved by countless number of methods.

For example, from your

Uxy=f(3x-2y)exp((2x+3y)/3)

it is follows immediately by double integration that the general solution can be in form

U(x,y)=F1(x)+F2(y)+\int \int (f(3x-2y)exp((2x+3y)/3)) dx dy .

My solution only looks simpler, but is equivalent the one above.
 

FAQ: How to Solve the PDE 2Uxxy + 3Uxyy - Uxy = 0?

What is a third order partial differential equation (PDE)?

A third order PDE is a mathematical equation that involves partial derivatives of a function with respect to three independent variables. It usually describes the relationship between a function and its three-dimensional inputs, and is commonly used in physics, engineering, and other scientific fields.

How do you solve a third order PDE?

Solving a third order PDE involves finding a function that satisfies the equation. This can be done using various methods such as separation of variables, substitution, and numerical methods. It may also involve using boundary conditions and initial conditions to determine the solution.

What are some applications of third order PDEs?

Third order PDEs have many applications in physics, engineering, and other fields. They are commonly used to describe phenomena such as heat transfer, fluid flow, and electromagnetism. They are also used in mathematical models for predicting and analyzing real-world systems.

Are there any special techniques for solving third order PDEs?

Some third order PDEs may have special techniques for solving them, such as the method of characteristics or the method of Frobenius. These techniques can be useful in certain cases, but may not always be applicable.

Can computer software be used to solve third order PDEs?

Yes, there are many computer software programs that can solve third order PDEs. These programs use numerical methods to approximate the solution, and can handle complex equations and boundary conditions. However, it is important to verify the accuracy of the solution obtained from the software.

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