M-274 Quiz: Montana State MATH 274 quiz2p3

dy/dx= (x^2-y^2)/(3xy)

The answer is supposed to be (x^2 - 4y^2)^3(x^2 )= C

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Here is what I have done:
x/(3y) - y/(3x) = (1/3)(x/y - y/x) = (1/3)(1/(y/x) - y/x)

v = y/x

dy/dx = v + x dy/dx

xdv/dx = 1/(3v) - 2v/3

dv/dx = (1/x)(1/(3v) - 2v/3)

integral of 3v(1-2v^2)^(-1)dv = lnx + C

Find by implicit differentiation.

Sketch the region of integration for the following double integral and change the order of integration. Double integral f(x,y) dy dx; where dx from 1 to 2 and dy from 0 to lnx