Find second derivative (d^2y/dx^2) of (-2x^2 + 4xy = 2x/y)

-2x^2 + 4xy = 2x/y

-4x + 4y + 4xy' = 2/y - 2x/y^2 y'
y' = (2xy^2-2y^3+y)/(2xy^2+x)
Now, using the quotient rule, we get
y" = [(2xy^2+4xyy'-6y^2y'+y')(2xy^2+x)-(2xy^2-2y^3+y)(2y^2+4xyy'+1)]/(2xy^2+x)^2

Substituting in for all the y' we get
y" = [(2xy^2+4xy((2xy^2-2y^3+y)/(2xy^2+x))-6y^2((2xy^2-2y^3+y)/(2xy^2+x))+((2xy^2-2y^3+y)/(2xy^2+x)))(2xy^2+x)-(2xy^2-2y^3+y)(2y^2+4xy((2xy^2-2y^3+y)/(2xy^2+x))+1)]/(2xy^2+x)^2

There's no elegant way to simplify this, but one possible result is

2y^2/[x^2(2y^2+1)^3] * [(4y^4+4y^2+4y+1)x^2 - 2(4y^4-8y^2-1)x + 2y(4y^4+4y^2-3)]

To find the second derivative (d^2y/dx^2) of the given equation (-2x^2 + 4xy = 2x/y), we need to follow these steps:

Step 1: Rewrite the equation in terms of y.
-2x^2 + 4xy = 2x/y
Multiply both sides by y to get rid of the denominator:
-2x^2y + 4xy^2 = 2x

Step 2: Differentiate both sides of the equation with respect to x.
Differentiating the left side involves using the product rule, while the right side is straightforward:
d/dx(-2x^2y) + d/dx(4xy^2) = d/dx(2x)

Using the product rule for each term on the left side:
[-2x^2 * (dy/dx) + (-2y) * (2x)] + [4x * (2y * (dy/dx)) + 4y^2] = 2

Simplifying the equation further:
-2x^2(dy/dx) - 4xy + 8xy(dy/dx) + 4y^2 = 2

Step 3: Rearrange the equation to solve for (dy/dx), the first derivative.
Group the terms that contain (dy/dx) and those that don't:
(-2x^2 + 8xy)(dy/dx) = 4xy - 4y^2 + 2

Divide through by (-2x^2 + 8xy) to isolate (dy/dx):
dy/dx = (4xy - 4y^2 + 2) / (-2x^2 + 8xy)

Step 4: Differentiate (dy/dx) obtained in Step 3 with respect to x to find the second derivative (d^2y/dx^2).
Differentiating both sides again using the quotient rule:
d/dx(dy/dx) = d/dx[(4xy - 4y^2 + 2) / (-2x^2 + 8xy)]

Using the quotient rule, we have:
[d/dx(4xy - 4y^2 + 2) * (-2x^2 + 8xy) - (4xy - 4y^2 + 2) * d/dx(-2x^2 + 8xy)] / (-2x^2 + 8xy)^2

Simplifying further, we get:
(4y - 8xy + 8y(dy/dx)) * (-2x^2 + 8xy) - (4xy - 4y^2 + 2) * (-4x + 8y(dy/dx))) / (-2x^2 + 8xy)^2

Simplifying more, we have:
(4y - 8xy + 8y * (dy/dx)) * (-2x^2 + 8xy) - (4xy - 4y^2 + 2) * (-4x + 8y * (dy/dx))) / (-2x^2 + 8xy)^2

This equation represents the second derivative (d^2y/dx^2) of the given equation (-2x^2 + 4xy = 2x/y).