# Method of characteristics

• gtfitzpatrick
In summary, the problem involves solving for u(x,y) using the method of characteristics. The first part of the solution involves finding the projected characteristic by setting ∂y/∂x = x/y, which results in y^2 + x^2 = k. However, the second part is proving to be difficult, as setting ∂u/∂x = 1/y does not yield the correct solution. Any suggestions for solving the second part would be appreciated.

## Homework Statement

solve using method of characteristics
$$y \frac{ \partial u}{ \partial x} - x \frac{ \partial u}{ \partial y}= 1$$ where u(x,0) = 0 for 0<x<infinity

## The Attempt at a Solution

$$\frac{ \partial y}{ \partial x} = \frac{x}{y}$$ which gives $$y^2+x^2= k$$ the projected characteristic. but its the second part that is giving me trouble when i go to find $$\frac{ \partial u}{ \partial x} = \frac{1}{y}$$ which doesn't work out right...any suggestion anyone?

hi gtfitzpatrick!

(have a curly d: ∂ )

∂y/∂x = x/y gives y2 minus x2 = k

tiny-tim said:
hi gtfitzpatrick!

(have a curly d: ∂ )

∂y/∂x = x/y gives y2 minus x2 = k

my mistake it should have read [/SIZE]

∂y/∂x = -x/y gives y2 + x2 = k

but don't know how to work the second part?

oh and hi back

## 1. What is the method of characteristics?

The method of characteristics is a mathematical technique used to solve partial differential equations. It involves constructing characteristic curves and using them to transform the partial differential equation into a set of ordinary differential equations, which can then be solved using standard techniques.

## 2. When is the method of characteristics used?

The method of characteristics is typically used to solve hyperbolic partial differential equations, such as those found in fluid dynamics or electromagnetism. It is also often used to solve initial value problems and boundary value problems.

## 3. How does the method of characteristics work?

The method of characteristics works by finding a set of curves, called characteristic curves, that are tangent to the solution of the partial differential equation at every point. These curves are then used to transform the original equation into a set of ordinary differential equations, which can be solved to find the solution.

## 4. What are the advantages of using the method of characteristics?

The method of characteristics has several advantages, including its ability to handle discontinuities and its ability to provide a visual representation of the solution. It is also typically more accurate and efficient than numerical methods for solving partial differential equations.

## 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 partial differential equations, it does have some limitations. It may not be applicable to all types of equations, and it can become computationally expensive for complex problems. Additionally, the method may yield multiple solutions or no solution at all in certain cases.