PDE separation of variable problem

So I've added it in and now the solution is correctIn summary, the conversation discusses solving a PDE using the method of separation of variables. The solution is found to be u(x,y)=5ye^{y-1+x^2-x}, with an update mentioning the inclusion of an arbitrary integration constant in the solution.
  • #1
JamesGoh
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0

Homework Statement



Using the method of separation of variables, obtain a solution of the following PDE subject to
the given conditions

[itex]\frac{du}{dx}+y\frac{du}{dy}=(2x+y)u[/itex]

[itex]u(x,1)=5e^{x^{2}-x}[/itex]


Homework Equations





The Attempt at a Solution



See my attached working

the answer is [itex]u(x,y)=5ye^{y-1+x^{2}-x}[/itex]

Im not sure if I am getting anywhere near close in my working out. Have double checked the arithemetic and everything seems fine so far
 

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  • #2
hi guys

an update on the problem. I managed to solve it.

The issue was that I didn't consider the arbirtrary integration constant, C, when doing the integral of the Y term.
 

What is PDE separation of variable problem?

PDE separation of variable problem is a method used to solve partial differential equations (PDEs) by representing the solution as a product of one function of each of the independent variables. This allows the PDE to be transformed into a set of ordinary differential equations (ODEs) which can then be solved using traditional methods.

What types of PDEs can be solved using separation of variables?

PDE separation of variable problem is usually used to solve linear PDEs with homogeneous boundary conditions. Examples of PDEs that can be solved using this method include the heat equation, wave equation, and Laplace's equation.

What are the steps involved in solving a PDE using separation of variables?

The first step is to express the PDE in terms of the independent variables and their derivatives. Then, the solution is assumed to be a product of functions of each independent variable. This is substituted into the PDE, which results in a set of ODEs. These ODEs can then be solved using boundary conditions and other techniques such as integration and differentiation.

What are some advantages of using separation of variables to solve PDEs?

One advantage is that it can be used to solve a wide range of linear PDEs. It also simplifies the problem by transforming it into a set of ODEs, which can then be solved using well-known techniques. Additionally, the solution obtained using separation of variables is often in a simple and interpretable form.

What are some limitations of using separation of variables to solve PDEs?

This method is only applicable to linear PDEs with homogeneous boundary conditions. It also requires the PDE to be expressed in terms of the independent variables and their derivatives, which may not always be possible. Additionally, the solution obtained using separation of variables may not be valid for all values of the independent variables, and may need to be checked for convergence and uniqueness.

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