Having Trouble Simplifying

[alabama14]

Determine whether the given relation is an implicit solution to the given differential equation. Assume that the relationship does define y implicitly as a function of x and use implicit differentiation.

e^(xy) + y = x - 1, dy/dx = (e^(-xy) - y)/(e^(-xy) + x)

I have gotten this far:
ye^(xy)(dy/dx) + dy/dx = 1
(dy/dx)(ye^(xy) + 1) = 1
dy/dx = 1/(ye^(xy) + 1)

How do I simplify this?

 

[Deleted member 4993]

Determine whether the given relation is an implicit solution to the given differential equation. Assume that the relationship does define y implicitly as a function of x and use implicit differentiation.

e^(xy) + y = x - 1, dy/dx = (e^(-xy) - y)/(e^(-xy) + x)

I have gotten this far:
ye^(xy)(dy/dx) + dy/dx = 1
(dy/dx)(ye^(xy) + 1) = 1
dy/dx = 1/(ye^(xy) + 1)

How do I simplify this?

\(\displaystyle \dfrac{d}{dx}\left [(y*e^{x*y}) \ + y\right ] \ \ = \dfrac{d}{dx}(\ x \ - \ 1)\)

\(\displaystyle \left [y'*e^{x*y} + y*e^{x*y}*(y + x*y')\right ] \ + \ y' \ = \ 1\)

Now solve for y' and continue.....

 

[tektonik]

just a minor careless error

Determine whether the given relation is an implicit solution to the given differential equation. Assume that the relationship does define y implicitly as a function of x and use implicit differentiation.

e^(xy) + y = x - 1, dy/dx = (e^(-xy) - y)/(e^(-xy) + x)

I have gotten this far:
ye^(xy)(dy/dx) + dy/dx = 1
(dy/dx)(ye^(xy) + 1) = 1
dy/dx = 1/(ye^(xy) + 1)

How do I simplify this?

For your first line of work, you should have this:

(x*dy/dx + y)e^(xy)(dy/dx) + dy/dx = 1

Not this:

ye^(xy)(dy/dx) + dy/dx = 1

The reason is that you should use the chain rule then the product rule to take the derivative of y with respect to x of e^xy:

dy/dx [e^xy] = (x*dy/dx + y)e^(xy)

You accidentally took the partial derivative of xy with respect to x:

del-y/del-x [xy] = y

I made the same exact mistake as you did initially lol. You can take it from there...