First order pde cauchy problem by method of characteristics

In summary, the conversation discusses a problem involving characteristic equations and how to solve them. The first and third equations are solved to get an expression for c_1, while the first and second equations yield an expression for c_2. These two expressions can be combined to find an arbitrary function f. The conversation also mentions the relationship between inverse tangent and natural logarithm.
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pk415
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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]


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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What is a first order PDE Cauchy problem?

A first order PDE Cauchy problem is a type of partial differential equation (PDE) that involves one independent variable and one dependent variable. It also includes initial conditions, known as the Cauchy data, which are used to determine a unique solution.

What is the method of characteristics?

The method of characteristics is a technique used to solve first order PDEs. It involves finding a set of curves, known as the characteristic curves, along which the solution of the PDE is constant. By using these curves, the PDE can be reduced to a system of ordinary differential equations, which can then be solved using standard methods.

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, follow these steps:

  1. Identify the characteristic curves by setting the PDE equal to a constant.
  2. Write down the equations for the characteristic curves.
  3. Use the initial conditions to determine the values of the constants in the equations.
  4. Solve the system of equations to obtain the solution to the PDE.

What are the advantages of using the method of characteristics to solve PDEs?

The method of characteristics has several advantages:

  • It can be applied to a wide range of first order PDEs.
  • It is a systematic and step-by-step approach to solving PDEs.
  • It allows for the use of initial conditions to determine a unique solution.
  • It can be used to solve PDEs with complex boundary conditions.

Are there any limitations to using the method of characteristics?

While the method of characteristics is a powerful tool for solving first order PDEs, it does have some limitations:

  • It is only applicable to first order PDEs.
  • The characteristic curves may not intersect, making it impossible to solve the PDE using this method.
  • It may not be able to handle PDEs with discontinuous or singular coefficients.

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