First Order PDE Cauchy problem Using Method of Characteristics

In summary, the conversation discusses solving a problem involving characteristic equations and finding an arbitrary function f. The function f is derived from the given Cauchy data and there is a relation between inverse tangent and natural logarithm.
  • #1
pk415
5
0

Homework Statement


Ok, so I can get through most of this but I can't seem to get the last part... Here is the problem

[tex]xU_x + (y^2+1)U_y = U-1; U(x,x) = e^x[/tex]


Homework Equations





The Attempt at a Solution



Characteristic equations are:

[tex]\frac{dx}{x} = \frac{dy}{y^2+1} = \frac{dU}{U-1}[/tex]

Solving the first and third gives:

[tex]\frac{U-1}{x} = c_1[/tex]

The first and second equation yield:

[tex]tan^{-1}(y) - lnx = c_2[/tex]

Put the two together in the form

[tex]c_1 = f(c_2)[/tex]

[tex]\frac{U-1}{x} = f(tan^{-1}(y) - lnx)[/tex]

Sub in the Cauchy data and you get

[tex]\frac{e^x-1}{x} = f(tan^{-1}(x) - lnx)[/tex]

Now how do I find what my arbitrary function f is? I have spent hours on this. Is there something that relates inverse tan to natural log? Arrggghhhh!

Thanks for any help.
 
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  • #2
Where did the function f come from? The way I see it, you end up with three equations:

[tex]\begin{cases}
x=C_1 e^{\tan^{-1}y}\\
U=C_2 x+1\\
U=C_3 e^{\tan^{-1}y}+1
\end{cases}[/tex]

Reconciling those, you get
[tex]U=Cxe^{\tan^{-1}y}+1[/tex]



Oh, and there is a relation between inverse tan and natural log: http://functions.wolfram.com/ElementaryFunctions/ArcTan/02/0001/

[tex]\tan^{-1}z={i \over 2}\ln{1-iz \over 1+iz}[/tex]
 

1. What is a First Order PDE Cauchy problem?

A First Order PDE Cauchy problem is a type of differential equation that involves a first order partial derivative of an unknown function with respect to multiple independent variables. The Cauchy problem is a specific type of initial value problem in which the unknown function is specified at a given point in the domain.

2. What is the Method of Characteristics?

The Method of Characteristics is a technique used to solve certain types of partial differential equations, specifically first order linear equations. It involves identifying characteristic curves within the solution domain and using them to determine the solution to the PDE.

3. How do you solve a First Order PDE Cauchy problem using the Method of Characteristics?

To solve a First Order PDE Cauchy problem using the Method of Characteristics, you first need to identify the characteristic curves within the solution domain. Then, using these curves, you can determine the solution to the PDE by setting up a system of equations and solving for the unknown function.

4. What are the advantages of using the Method of Characteristics?

The Method of Characteristics is advantageous because it allows for the solution of certain types of PDEs that cannot be solved using other methods. It also provides a systematic approach to solving these types of equations and can be applied to a wide range of physical problems.

5. Are there any limitations to using the Method of Characteristics?

While the Method of Characteristics is a powerful tool for solving certain types of PDEs, it does have limitations. It can only be applied to linear equations and may not work for more complex equations. Additionally, it may be difficult to identify characteristic curves in some cases, making the method less applicable.

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