find general solution in implicit form of differential equation dy/dx= 2(e^x-e^-x)/y^2(e^x+e^-x)^4 (y>0)

To find the general solution of the given differential equation, we will first separate the variables and then integrate both sides of the equation.

The given differential equation is:
dy/dx = 2(e^x - e^(-x)) / [y^2(e^x + e^(-x))^4]

To separate the variables, we multiply both sides of the equation by [y^2(e^x + e^(-x))^4]:
[y^2(e^x + e^(-x))^4] dy = 2(e^x - e^(-x)) dx

Now, we integrate both sides of the equation:
∫[y^2(e^x + e^(-x))^4] dy = ∫2(e^x - e^(-x)) dx

The integral on the left side can be simplified using substitution. Let's substitute u = e^x + e^(-x):

To find du, we take the derivative of u with respect to x:
du/dx = d/dx(e^x + e^(-x)) = e^x - e^(-x)

Rearranging the equation, we get du = (e^x - e^(-x)) dx

Substituting these values into the integral on the left side:
∫[y^2u^4] dy = ∫2du

Integrating both sides separately:
(y^2/5)u^5 = 2u + C

Substituting the value of u back:
(y^2/5)(e^x + e^(-x))^5 = 2(e^x + e^(-x)) + C

Simplifying the equation:
(y^2/5)(e^5x + 5e^3x + 10e^x + 10e^(-x) + 5e^(-3x) + e^(-5x)) = 2(e^x + e^(-x)) + C

This is the general solution to the given differential equation in implicit form.