Method of characteristics for a 1st order quasi-linear PDE.

[sapiental]

Hi,

I'm looking over the examples in my book for this problem and the general approach is

a(x,y,z)*u_x + b(x,y,z)*u_y = c(x,y,z)

where u(x,y)

I have the following problem in my notes:

1/x * u_x + 1/y * u_y = x^2 * sqrt(z)

and I get the solution easily because of the format:

Another problem i did was

y*u_x - x * u_y = 2xyu


where i multiplied both sides by 1/xy to get:

1/x*u_x - 1/y * u_y = 2u


the solution followed easily because of the classic format:

however, what i don't get is this format

u*u_x + y*u_y = x

the problem is an ivp with the following characteristics:

x = s, y = s, u = 2s

z = u(x_o,y_o) = u(s,s) = 2s

how do i get the solutin when it's not in the standard format? Or am i misinterpreting

a(x,y,z)*u_x + b(x,y,z)*u_y = c(x,y,z)

in the sense that the coefficient functions (a(x,y,z), b(x,y,z), c(x.y,z) can be any variable x,y,u(x_o,y_o)?

I don't have the book and all my examples are written in the convenient form I'm after which confuses me

thank you

 

[jvicens]

Thank you for your question. The format you are referring to is known as the "general form" or "canonical form" of a first-order partial differential equation. It is often used in textbooks as it is a standardized way of representing a wide range of equations. However, it is important to note that not all equations can be easily transformed into this form and may require different techniques for solving.

In your first example, you correctly identified the coefficients a(x,y,z), b(x,y,z), and c(x,y,z) and were able to solve the equation using the standard approach. In your second example, you multiplied both sides by 1/xy to transform it into the standard form, which allowed for an easy solution. However, in your third example, the equation is not in the standard form and therefore cannot be solved using the same approach.

In this case, it may be helpful to use a different technique, such as separation of variables or the method of characteristics, to solve the equation. It is also possible that the equation cannot be solved analytically and may require numerical methods.

In summary, while the standard form is a useful and common format for representing first-order partial differential equations, it is not the only approach and some equations may require different techniques for solving. I hope this helps to clarify your confusion. If you have any further questions, please don't hesitate to ask.