Find in implicit form the general solution of differential equation dy/dx=2(e^x-e^-x)/y^2(e^x+e^-x)^4 with (y>0). I know this requires a seperation of variables but I beyond that I am confused by how.Thanks.

Answer 1

dy/dx=2(first )/y^2 (second)

y^2 dy=2(first)dx/(second)

so lets look at the right side.

2(e^x-e^-x)/(e^x+e^-x)^4

factor out e^-x

numerator: 2e^-x(e^2x)

denominator(factor out e^-4x)
e^-4x (e^2x-1)^2

so you are left on the right side:

2*e^3x(e^2x -1)/(e^2x+1) dx

or = 2(e^5x)/(e^2x+1) dx + 2e^3x/(e^2x+2) dx

Answer 2

thanks Bob much appreciated. Do I then take the integral of the last line? And is the left hand side 1/3y^3 (from y^)? I do not know how this fits together. Thanks for any further help