Answer to Question #220558 in Differential Equations for Al Amin

Question

 Eliminate 𝑎, 𝑏 and 𝑐 from 𝑧 = 𝑎(𝑥 + 𝑦) + 𝑏(𝑥 + 𝑦) + 𝑎𝑏𝑡 + 𝑐.

Answer 1

Eliminate 𝑎, 𝑏 and 𝑐 from $𝑧 = 𝑎(𝑥 + 𝑦) + 𝑏(𝑥 - 𝑦) + 𝑎𝑏𝑡 + 𝑐.$

Differentiate the given equation partially with respect to $x, y,$ and $t$

$$\dfrac{\partial z}{\partial x}=a+b, \dfrac{\partial z}{\partial y}=a-b, \dfrac{\partial z}{\partial t}=ab$$

$$(a+b)^2-(a-b)^2=a^2+2ab+b^2-a^2+2ab-b^2=4ab$$

Then

$$(\dfrac{\partial z}{\partial x})^2-(\dfrac{\partial z}{\partial y})^2=4\dfrac{\partial z}{\partial t}$$

This is a partial differential equation of order one and degree two.