Final answer:
To find y'' at x = 0, first differentiate the given equation twice with respect to x. Then substitute x = 0 into the equation to find the value of y''.
Step-by-step explanation:
To find the value of y'' at the point where x = 0, we need to differentiate the equation xy = e^y = e twice with respect to x. Let's start by differentiating both sides of the equation with respect to x:
1. Differentiating xy = e^y = e:
y + xy' = e^y * y' = 0
2. Differentiating again:
(y' + xy'') + x(y'' + y'^2) = 0
Now, substitute x = 0 into the equation:
y'' = -y'^2/y
Since x = 0 is given, we can substitute it into y = e^y = e:
e = e^y where y = 1
Therefore, when x = 0, y'' = -y'^2/y = -1^2/1 = -1