Question

Verify that the pfaffian differential equation yz dx+ (x^2y-zx) dy +(x^2z-xy) dz = 0 is integrable and hence find its integral.

Answer 1

Answer on Question #66333 – Math – Differential Equations

Question

Verify that the pfaffian differential equation $yz \, dx + (x^2 y - zx) dy + (x^2 z - xy) dz = 0$ is integrable and hence find its integral.

Solution

We have

$$y z d x + (x ^ {2} y - z x) d y + (x ^ {2} z - x y) d z = 0$$

General form of the Pfaffian equation is

$$P d x + Q d y + R d z = 0$$

The integrability condition for the Pfaffian equation is [1, page 384]

$$(\operatorname {c u r l} F, F) = 0$$

where $F = (P,Q,R)$, or

$$P \left(\frac {\partial Q}{\partial z} - \frac {\partial R}{\partial y}\right) + Q \left(\frac {\partial R}{\partial x} - \frac {\partial P}{\partial z}\right) + R \left(\frac {\partial P}{\partial y} - \frac {\partial Q}{\partial x}\right) = 0$$

Verify this condition for the given equation. We get

$$\begin{array}{l} P \left(\frac {\partial Q}{\partial z} - \frac {\partial R}{\partial y}\right) + Q \left(\frac {\partial R}{\partial x} - \frac {\partial P}{\partial z}\right) + R \left(\frac {\partial P}{\partial y} - \frac {\partial Q}{\partial x}\right) = \\ = y z (- x + x) + (x ^ {2} y - z x) (2 x z - y - y) + (x ^ {2} z - x y) (z - 2 x y + z) = \\ = 2 x (x y - z) (x z - y) + x (x z - y) 2 (z - x y) = 0 \\ \end{array}$$

The integrability condition for this equation hold.

If the Pfaffian equation is multiplied by a certain function $\mu(x, y, z)$ then one can obtain in the left-hand side the total differential

$$\frac {\partial u}{\partial x} d x + \frac {\partial u}{\partial y} d y + \frac {\partial u}{\partial z} d z = d u = 0$$

That gives the solution of the Pfaffian equation $u = \text{const}$

multiply the original equation by $\frac{1}{x^2}$. We get

$$\begin{array}{l} \frac {y z}{x ^ {2}} d x + \left(y - \frac {z}{x}\right) d y + \left(z - \frac {y}{x}\right) d z = 0 \\ \left(\frac {y z}{x ^ {2}} d x - \frac {z}{x} d y - \frac {y}{x} d z\right) + y d y + z d z \\ d \left(- \frac {y z}{x}\right) + d \left(\frac {y ^ {2}}{2}\right) + d \left(\frac {z ^ {2}}{2}\right) = 0 \\ d \left(- \frac {y z}{x} + \frac {y ^ {2}}{2} + \frac {z ^ {2}}{2}\right) = 0 \\ \end{array}$$

Finally we get solution

$$- \frac {y z}{x} + \frac {y ^ {2}}{2} + \frac {z ^ {2}}{2} = C$$

**Answer**: The differential equation is integrable. The integral of the original equation is

$$- \frac {y z}{x} + \frac {y ^ {2}}{2} + \frac {z ^ {2}}{2} = C$$

Reference:

[1] Daniel Zwillinger. Handbook of Differential Equations, 3rd edition

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