Can Implicit Differentiation Reconstruct the Original Equation?

[shreddinglicks]

Homework Statement

This is just something I am doing for fun. If you see my attachment you will see I took an equation (underlined) and differentiated it implicitly to give me a separable equation (boxed)

How do I solve the separable equation to work backwards to get the underlined equation?

I apologize in advance, I forgot to bring the negative sign into the separable equation.

Homework Equations

in attachment

The Attempt at a Solution

in attachment

 

[Mark44]

Homework Statement

This is just something I am doing for fun. If you see my attachment you will see I took an equation (underlined) and differentiated it implicitly to give me a separable equation (boxed)

How do I solve the separable equation to work backwards to get the underlined equation?

I apologize in advance, I forgot to bring the negative sign into the separable equation.

Homework Equations

in attachment

The Attempt at a Solution

in attachment

From your attachment, here's what you have:
$y^2 + 2xy = C$
$2y\frac{dy}{dx} + 2y + 2x\frac{dy}{dx} = 0$
$\frac{dy}{dx} 2x + 2y = -2y$
$\frac{dy}{dx} = \frac{-y}{x + y}$ (added the minus sign that you mentioned)

The third line needs parentheses around 2x + 2y.
Your final equation is not separable. Another tactic must be used. One that works is to let u = y/x. That will give you a separable DE.