py + qx = pq

[Гост]

[tex]py + qx = pq, p=\frac{\partial z}{\partial x},q=\frac{\partial z}{\partial y}[/tex]

[Гост]

[tex](p – x) (q – y) = pq – py – qx + xy\Rightarrow pq – py – qx + xy = xy\Rightarrow (p – x) (q – y) = xy\Rightarrow (p – x) / (x) = (y) / (q – y)[/tex]
[tex]\Rightarrow (p – x) / (x) = a = (y) / (q – y)[/tex]
[tex](p – x) / (x) = a\Rightarrow p = (a + 1)x\Rightarrow ∫ z_x dx = (a + 1) ∫ x dx + f(y)\Rightarrow z = (a + 1) (x^2 / 2) + f(y)[/tex]
[tex](y) / (q – y) = a\Rightarrow q = ((a + 1) / a)y\Rightarrow ∫ z_y dy = ((a + 1) / (a)) ∫ y dy + g(x)\Rightarrow z = ((a +1) / (a)) (y^2 / 2) + g(x)[/tex]
[tex]\Rightarrow z = (a + 1) ((x^2 / 2) + (y^2 / 2a)) + b[/tex]
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