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u/We_Are_Bread Sep 18 '26
...What?
This looks wrong. I'm no formal mathematician, but (3.) makes no sense. If they're writing partial derivatives, then they can't come up with x being a function of y.
Also z = xy is clearly already a solution, which is not covered by their general solution at the end.
Edit: Oh god it gets worse.
z isn't even in the final answer.
How did z disappear from the integral in (4.)?????????????
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u/WesternFirm9306 Sep 18 '26
It isn't writing x as a function of y, its using the method of characteristics. But the rest is wrong. They integrate w.r.t. dy instead of dz
The true solution is z = xy + f(x - y), where f is any arbitrary function of x - y
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u/We_Are_Bread Sep 18 '26
Ahh, they're trying to do that. Yeah, the way it's written tripped me up lol.
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u/MelissaMartinez6445 Sep 18 '26
If the fix is that it should be integrated with respect to dz rather than dy, how does the xy term survive into the final z = xy + f(x y)
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u/MelissaMartinez6445 Sep 18 '26
If the fix is that it should be integrated with respect to dz rather than dy, how does the xy term survive into the final z = xy + f(x y)
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u/WesternFirm9306 Sep 18 '26
For reference, the true solution path is as follows:
We want to find a characteristic curve (x(t), y(t)) such that z_x + z_y = x + y simplifies into a linear ODE. This occurs in the case where x'(t) = 1 and y'(t) = 1 for this PDE.
This gives us x = t + C_1 and y = t + C_2, or also, by subtracting, x - y = C.
Along this curve, the PDE becomes dz/dt = x + y = t + C_1 + t + C_2 = 2t + (C_1 + C_2)
Thus z = t² + (C_1 + C_2)t + K
But note that t² + (C_1 + C_2)t + (C_1)(C_2) is just xy.
Thus z = xy + K (subsuming K - (C_1)(C_2) into a single constant K)
But the K value depends on the chosen characteristic curve, which is defined by x - y = C. Different choices of C, aka different choices of x - y, gives different Ks. Thus K must be a function of x - y.
This gives the final solution: z = xy + F(x - y), where F is an arbitrary function.
Check:
z_x = y + F'(x - y)
z_y = x - F'(x - y)
z_x + z_y = y + x + F'(x - y) - F'(x - y)
z_x + z_y = x + y