Solve dydx x2 6y22xySolutionLet y vxThen dydx v xdvdx a

Solution

Let y = vx.Then dy/dx = v + xdv/dx and (x² -6y²)/2xy = ½(x/y) -6/2( y/x) = 1/2v -3v. Then the given equation changes to v+xdv/dx = 1/2v -3v or, xdv/dx = 1/2v – 4v = (1- 8v²)/2v so that 2vdv/(1- 8v²) = dx/x. Then, 2vdv/(1- 8v2) = dx/x. Now, let (1-8v²) = u. Then -16vdv = du so that 2vdv/(1- 8v²) = -du/8u so that dx/x = -1/8 du/u. Then -1/8 ln I u I = ln I x I + c,( where c is a constant) or, -1/8 ln I 1-8v² I = ln IxI + c or, -1/8 ln I 1- 8y²/x² I = ln I x I + c.