Question

find the complete integral of the equation (dz/dx1)(dz/dx2)(dz/dx3)= (z^3 X1 X2 X3)

Answer 1

Answer to Question #74060, Math / Differential Equations

Find the complete integral of the equation

$$\frac {\partial z}{\partial x _ {1}} \cdot \frac {\partial z}{\partial x _ {2}} \cdot \frac {\partial z}{\partial x _ {3}} = z ^ {3} x _ {1} x _ {2} x _ {3}$$

Solution.

$$\left(\frac {1}{z} \frac {\partial z}{\partial x _ {1}}\right) \left(\frac {1}{z} \frac {\partial z}{\partial x _ {2}}\right) \left(\frac {1}{z} \frac {\partial z}{\partial x _ {3}}\right) = x _ {1} x _ {2} x _ {3}$$

Let

$$\frac {1}{z} d z = d Z \Rightarrow \ln z = Z$$

Then:

$$\left(\frac {\partial Z}{\partial x _ {1}}\right) \left(\frac {\partial Z}{\partial x _ {2}}\right) \left(\frac {\partial Z}{\partial x _ {3}}\right) = x _ {1} x _ {2} x _ {3}$$ $$P _ {1} P _ {2} P _ {3} = x _ {1} x _ {2} x _ {3}$$ $$P _ {1} P _ {2} P _ {3} - x _ {1} x _ {2} x _ {3} = 0$$ $$f \left(x _ {1}, x _ {2}, x _ {3}, P _ {1}, P _ {2}, P _ {3}\right) = P _ {1} P _ {2} P _ {3} - x _ {1} x _ {2} x _ {3} = 0$$

Jacobi's equations are

$$\frac {d P _ {1}}{\partial f / \partial x _ {1}} = \frac {d x _ {1}}{- \partial f / \partial P _ {1}} = \frac {d P _ {2}}{\partial f / \partial x _ {2}} = \frac {d x _ {2}}{- \partial f / \partial P _ {2}} = \frac {d P _ {3}}{\partial f / \partial x _ {3}} = \frac {d x _ {3}}{- \partial f / \partial P _ {3}}$$ $$\frac {d P _ {1}}{- x _ {2} x _ {3}} = \frac {d x _ {1}}{- P _ {2} P _ {3}} = \frac {d P _ {2}}{- x _ {1} x _ {3}} = \frac {d x _ {2}}{- P _ {1} P _ {3}} = \frac {d P _ {3}}{- x _ {1} x _ {2}} = \frac {d x _ {3}}{- P _ {1} P _ {2}}$$

Since

$$P _ {2} P _ {3} = \frac {x _ {1} x _ {2} x _ {3}}{P _ {1}}$$

Then:

$$\frac {d P _ {1}}{- x _ {2} x _ {3}} = \frac {d x _ {1}}{- (x _ {1} x _ {2} x _ {3} / P _ {1})}$$

Answer to Question #74060, Math / Differential Equations

$$\frac {d P _ {1}}{P _ {1}} = \frac {d x _ {1}}{x _ {1}} \Rightarrow \ln P _ {1} = \ln x _ {1} + \ln a _ {1}$$

Thus:

$$P _ {1} = a _ {1} x _ {1}$$ $$P _ {2} = a _ {2} x _ {2}$$ $$P _ {3} = x _ {3} / a _ {1} a _ {2}$$ $$d Z = P _ {1} d x _ {1} + P _ {2} d x _ {2} + P _ {3} d x _ {3}$$ $$d Z = a _ {1} x _ {1} d x _ {1} + a _ {2} x _ {2} d x _ {2} + \frac {x _ {3}}{a _ {1} a _ {2}} d x _ {3}$$ $$Z = \ln z = \frac {a _ {1} x _ {1} ^ {2}}{2} + \frac {a _ {2} x _ {2} ^ {2}}{2} + \frac {x _ {3} ^ {2}}{2 a _ {1} a _ {2}} + \frac {a _ {3}}{2}$$

Answer:

$$2 \ln z = a _ {1} x _ {1} ^ {2} + a _ {2} x _ {2} ^ {2} + \frac {x _ {3} ^ {2}}{a _ {1} a _ {2}} + a _ {3}$$

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