Solving 1st Order PDE with Initial Condition - Help Needed

In summary, the conversation discusses solving the equation Ux + Uy + U = e^-(x+y) with the initial condition U(x,0)=0. The speaker mentions that they found a solution that satisfies the equation but not the initial condition. They suggest that adding a factor of y might help, but they have not found a satisfactory equation yet. Another person offers a hint to try using a substitution involving two variables, z and t.
  • #1
BustedBreaks
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0
I'm trying to solve this equation:

Ux + Uy + U = e^-(x+y) with the initial condition that U(x,0)=0


I played around and and quickly found that U = -e^-(x+y) solves the equation, but does not hold for the initial condition. For the initial condition to hold, I think there needs to be some factor of y in the equation for U, but after trying a few equations, I can't find one that satisfies both the initial condition and the equation.

Can someone throw me a hint?

Thanks!
 
Last edited:
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  • #2
You might try the substitution [itex]u(x,y)=v(x,y)e^{-(x+y)}[/itex]...
 
  • #3
You need to variables for the plane:

[tex]z=x+y, t=x-y [/tex]

your equation only depends on one of them

[tex] 2u_{,z}+u=e^{-z}[/tex]
 
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1. What is a first order partial differential equation (PDE)?

A first order PDE is a mathematical equation that involves partial derivatives of a function with respect to more than one independent variable. It can be used to describe the relationship between multiple quantities in a system.

2. What is the process for solving a first order PDE with an initial condition?

The first step is to identify the type of PDE and then use the appropriate method (such as separation of variables or method of characteristics) to solve it. The initial condition is then used to find the unique solution to the PDE.

3. What are some common techniques for solving first order PDEs?

Some common techniques include separation of variables, method of characteristics, and the Laplace transform. Each method has its own advantages and is useful for different types of PDEs.

4. What is the importance of initial conditions in solving PDEs?

The initial condition acts as a starting point for the solution of the PDE and helps to determine the unique solution. Without an initial condition, the solution may have multiple possible solutions.

5. How are first order PDEs used in science and engineering?

First order PDEs are used to describe many physical phenomena, such as heat transfer, fluid flow, and electrical circuits. They are also used in mathematical models to make predictions and solve real-world problems in various fields of science and engineering.

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