Question related to PDE y(z_x)+x(z_y)+z=y

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In summary, the conversation discusses solving a partial differential equation using the Method of Characteristics. The resulting equation involves variables x, y, and z, and a constant A which may be positive or negative. The question is raised about the necessity of accounting for a negative A in the solution function.
  • #1
coverband
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To solve the PDE:
y(z_x)+x(z_y)+z=y

Use Method of characteristics
a=y
b=x
d-cz=y-z

Thus
dx/y=dy/b=dz/(y-z)

Taking first and second term
xdx=ydy
x^2-y^2=A
x=sqrt(y^2+A)

My question is, at this stage of the calculation, must we account for a negative constant A such that
x=sqrt(y^2-A)
and if not why not?

Thank you
 
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  • #2
To be honest I'm not familier with the method.

Is it not that your x and y are the independent variables? So what is the domain for the solution function. A is just a constant. It can be positive or negative. I think!
 
  • #3
Anyone else!?
 

1. What is a PDE?

A PDE, or partial differential equation, is a type of mathematical equation that involves multiple variables and their partial derivatives. It is used to describe physical phenomena and is commonly used in fields such as physics, engineering, and economics.

2. What does the notation y(z_x)+x(z_y)+z=y mean?

This notation represents a specific type of PDE known as a first-order linear PDE. The variables y, z, and x are functions of two independent variables, and the equation shows how the partial derivatives of these functions interact with each other.

3. What is the significance of solving PDEs?

Solving PDEs allows us to understand and predict the behavior of complex systems in various fields. It is essential in developing mathematical models for real-world problems and has numerous applications in science and technology.

4. Can you provide an example of a physical system described by this PDE?

One example is the diffusion of a substance in a liquid or gas. The variable y could represent the concentration of the substance, while z and x represent the spatial coordinates. The equation would then describe how the concentration changes over time and space.

5. What are some common solution techniques for PDEs?

There are several techniques for solving PDEs, including separation of variables, the method of characteristics, and numerical methods. The choice of method depends on the specific PDE and its boundary conditions.

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