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

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The discussion focuses on solving the partial differential equation (PDE) y(z_x) + x(z_y) + z = y using the Method of Characteristics. The participants derive the relationship dx/y = dy/b = dz/(y-z) and explore the implications of the constant A in the equation x^2 - y^2 = A. The consensus is that A can be either positive or negative, as it represents a constant that does not restrict the independent variables x and y.

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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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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!
 
Anyone else!?
 

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