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physics604
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Homework Statement
$$xy+e^y=e$$ Find the value of y'' at the point where x=0.
Homework Equations
Chain rule
The Attempt at a Solution
I find y' first.
$$x\frac{dy}{dx}+y+e^y\frac{dy}{dx}=0$$ $$(x+e^y)\frac{dy}{dx}=-y$$ $$\frac{dy}{dx}=\frac{-y}{x+e^y}$$
Then I find y''.
$$\frac{(x+e^y)(-\frac{dy}{dx})-(-y)(1+e^y\frac{dy}{dx})}{(x+e^y)^2}$$ $$\frac{(x+e^y)(\frac{y}{x+e^y})-(-y-ye^y(\frac{-y}{x+e^y}))}{(x+e^y)^2}$$ $$\frac{y-(-y+\frac{y^2e^y}{x+e^y})}{(x+e^y)^2}$$ $$\frac{y+y-\frac{y^2e^y}{x+e^y}}{(x+e^y)^2}$$ $$(2y-\frac{y^2e^y}{x+e^y})(\frac{1}{(x+e^y)^2})$$ $$\frac{2y(x+e^y)-y^2e^y}{(x+e^y)^3}$$ $$\frac{2xy+2ye^y-y^2e^y}{(x+e^y)^3}$$
x=0 so plugging that in I get
$$\frac{ye^y(2-y)}{e^{3y}}=\frac{y(2-y)}{e^{2y}}$$
The answer is $$\frac{1}{e^2}$$ What did I do wrong?
Any help is much appreciated.
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