Solve by Method of Characteristics

[syj]

Homework Statement

$u\frac{\delta u}{\delta x}$+$\frac{\delta u}{\delta y}$ =1

$u|_{x=y}=\frac{x}{2}$

Homework Equations

the characteristic equations to solve are:

$\frac{dx}{ds}=u$

$\frac{dy}{ds}=1$

$\frac{du}{ds}=1$

The Attempt at a Solution

I got the following equations from the characteristic equations:

$\frac{dx}{ds}=u$ means $\frac{d^2x}{ds^2}=\frac{du}{ds}=1$
after integrating twice i get $x=\frac{1}{2}s^2+ks+x_0$

$y = s +y_{0}$

$u = s +u_{0}$

and here is where I am now stuck.
I also don't know how to apply the condition given : $u|_{x=y}=\frac{x}{2}$

 

[mighty2000]

Hello,

Thank you for your post. It seems that you are working on a partial differential equation (PDE) problem involving the variable u. In order to solve this problem, we first need to understand what the equation represents and what it is asking us to find.

From the given equation, it looks like we are dealing with a first-order linear PDE. This means that the highest derivative present is only first-order, and the equation is linear in u. This type of PDE can be solved using the method of characteristics, which is what you have started to do by finding the characteristic equations.

The characteristic equations are a set of differential equations that help us to find the solution to the PDE. They represent the curves along which the solution of the PDE is constant. In this case, we have three characteristic equations, one for each variable x, y, and u.

To solve the PDE, we need to find a solution that satisfies both the given equation and the initial condition u|_{x=y}=\frac{x}{2}. This initial condition tells us that at the points where x=y, the value of u is equal to half of x. This means that our solution must satisfy this condition at all points along the characteristic curves.

To find the solution, we can use the method of characteristics to find the general solution, and then use the initial condition to find the specific solution that satisfies both the PDE and the initial condition.

I hope this helps to clarify the problem and guide you in finding the solution. Keep up the good work in solving this PDE!