Partial Differential Equation

In summary, the conversation is about solving a partial differential equation using a separable solution and finding a general solution. The boundary condition and using a power series are also discussed. The expert summarizer provides a solution using series and constants, but the person asking for help is unable to understand and continues to ask for a complete working. The expert then becomes frustrated and leaves the conversation.
  • #1
EDerkatch
14
0
x(δu/δx)-(1/2)y(δu/δy)=0

By first looking for a separable solution of the form u(x, y)=X(x)Y(y), find the general solution of the equation given above.

Determine the u(x,y) which satisfies the boundary condition u(1,y)=1+siny

For the separable form I have u(x, y)=A(x^c)(y^2c), could someone please show me how to do the rest of it.

Thank you.
 
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  • #2
Write sin(y) as a power series.
 
  • #3
Each choice of c gives you a solution of your diff.eq.
Since your diff.eq is linear, a sum of such solutions is also a solution of your diff.eq.
 
  • #4
How do I get from u(x, y)=A(x^c)(y^2c) to the general solution?

Thank you.
 
  • #5
All right:
A series solution of your diff.eq is:
[tex]u(x,y)=\sum_{n=1}^{\infty}A_{n}x^{c_{n}}y^{2c_{n}}[/tex],
whereby follows:
[tex]u(1,y)=\sum_{n=1}^{\infty}A_{n}y^{2c_{n}}[/tex]
and [itex]A_{n},c_{n}[/itex] are constants.

Now, how can you fit this expression for u(1,y) to the given boundary condition?
 
  • #6
arildno said:
All right:
A series solution of your diff.eq is:
[tex]u(x,y)=\sum_{n=1}^{\infty}A_{n}x^{c_{n}}y^{2c_{n}}[/tex],
whereby follows:
[tex]u(1,y)=\sum_{n=1}^{\infty}A_{n}y^{2c_{n}}[/tex]
and [itex]A_{n},c_{n}[/itex] are constants.

Now, how can you fit this expression for u(1,y) to the given boundary condition?

Please go on...
 
  • #7
Well, use my first hint in post 2.
 
  • #8
arildno said:
Well, use my first hint in post 2.

Ok if you could please show me the COMPLETE working I would really appreciate it... Thank you.
 
  • #9
arildno said:
Well, use my first hint in post 2.

Ok if you could please show me the COMPLETE working I would really appreciate it... Thank you.
 
  • #10
Do you know what a power series is? :confused:
 
  • #11
arildno said:
Do you know what a power series is? :confused:

Yes lol, I just can't do this question, could you please show me the working for it... In fact can you do it yourself?
 
  • #12
If you are not capable of doing basic algebra, you should not be attempting partial differential equations!

(Yes, I can do it myself! That's not really the point is it? You have been told exactly HOW to solve your equation, yet you have not even TRIED to apply what you have been told.)
 
  • #13
EDerkatch said:
Yes lol, I just can't do this question, could you please show me the working for it... In fact can you do it yourself?

That's it. I'm out of here. It is long since I've met a more ungrateful and lazy f*ckhead on PF as you.
 

Related to Partial Differential Equation

1. What is a partial differential equation (PDE)?

A partial differential equation is a mathematical equation that involves partial derivatives of an unknown function with respect to multiple independent variables. It is used to describe physical phenomena that vary in space and time, such as heat transfer, fluid dynamics, and quantum mechanics.

2. What is the difference between a PDE and an ordinary differential equation (ODE)?

The main difference between a PDE and an ODE is that a PDE involves partial derivatives, while an ODE only involves ordinary derivatives. This means that a PDE describes systems that vary in multiple dimensions, while an ODE describes systems that vary in only one dimension.

3. What are the applications of PDEs in science and engineering?

PDEs have a wide range of applications in science and engineering, including modeling physical phenomena such as heat transfer, fluid flow, and electromagnetic fields. They are also used in financial mathematics, image processing, and computer graphics.

4. Are there different types of PDEs?

Yes, there are several types of PDEs, including elliptic, parabolic, and hyperbolic. These types differ in the behavior of their solutions and the type of initial and boundary conditions that are required to solve them.

5. How are PDEs solved?

There are several methods for solving PDEs, including analytical methods such as separation of variables and integral transforms, as well as numerical methods such as finite difference, finite element, and spectral methods. The appropriate method depends on the type of PDE and the specific problem being solved.

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