Did I do this right? (Differential Eqtns)

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Homework Statement


Show that the given equation is an implicit solution of the given differential equation:

Homework Equations



e(x - y) + e(y - x) (dy / dx) = 0

e2y + e2x = 1

The Attempt at a Solution



(e2y) = 1 - e2x

solve for dy/dx
dy/dx (e2y) = dy/dx (1 - e2x)

(2e2y)(dy / dx) = -2e2x

dy / dx = -e2x / e2y


Substitute: e(x - y) + { e(y - x) * [ -e2x / e2y ] } = 0

Now I am uncomfortable on my algebra here, and I try to muliply out the
{ e(y - x) * [ -e2x / e2y ] }
(please do not laugh if I am wrong) :

e(x - y) -[/color] e(y - x + 2x - 2y) = 0

e(x - y) - e(x - y) = 0
 
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I think you algebra is OK.

I don't like your notation for differentiation.

Does you class actually show differentiation on both sides of the equation as:
dy/dx (e2y) = dy/dx (1 - e2x) ?

I prefer something like

D_x ( e^{2y}) = D_x (1 - e^{2x})

or

\frac {d( e^{2y}) }{dx} = \frac { d(1-e^{2x})}{dx}


2 e^{y} \frac{dy}{dx} = -2 e^{2x}
 
To add to what Stephen Tashi said about your differentiation notation, dy/dx means the derivative of y with respect to x. An expression such as
dy/dx(e2y) means the derivative of y with respect to x times e2y. It does NOT mean that you are taking the derivative of e2y.

It's important to understand the difference between dy/dx(e2y) and d/dx(e2y). The first is a product of two functions; the second indicates an operation that hasn't been performed yet.
 
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