1. ## Rearranging

What does the rearranging lead to?

$\displaystyle \frac{\partial a}{\partial x} \partial y = \frac{\partial b}{\partial y} \partial x$

Dividing through by $\displaystyle \partial x$, which would it lead to?

(1)
$\displaystyle \frac{\partial a}{\partial x^2} \partial y = \frac{\partial b}{\partial y}$

(2)
$\displaystyle \frac{\partial ^2 a}{\partial x^2} \partial y = \frac{\partial b}{\partial y}$

Equation (1) or (2)?

2. Hello, Simplicity!

What does the rearranging lead to?

. . $\displaystyle \displaystyle \frac{\partial a}{\partial x} \partial y \:=\: \frac{\partial b}{\partial y} \partial x$

Dividing through by $\displaystyle \partial x$, which would it lead to?

$\displaystyle \displaystyle (1)\;\frac{\partial a}{\partial x^2} \partial y \:=\: \frac{\partial b}{\partial y}$
. . . . . . . or
$\displaystyle \displaystyle (2)\;\frac{\partial ^2 a}{\partial x^2} \partial y \:=\: \frac{\partial b}{\partial y}$

We have: .$\displaystyle \displaystyle \frac{\partial a}{\partial x}\partial y \;=\;\frac{\partial b}{\partial y}\partial x$

Dividing through by $\displaystyle \partial x$, we have: .$\displaystyle \displaystyle \frac{\partial a}{\partial x}\cdot \frac{\partial y}{\partial x} \;=\;\frac{\partial b}{\partial y}\cdot\frac{\partial x}{\partial x}$

Then we have: .$\displaystyle \displaystyle\frac{\partial a}{\partial x}\cdot\frac{\partial y}{\partial x} \;=\;\frac{\partial b}{\partial y}$

. . which can be written: .$\displaystyle \displaystyle \frac{\partial a\partial y}{\partial x^2} \;=\;\frac{\partial b}{\partial y}$ . . . answer (1)