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Solution :

On dividing the equation vy `y^(4)`, we have <br> `(1)/(y^(4)) (dy)/(dx) - ((2)/(y^(3)).(1)/(x)) = 1` <br> Set `(1)/(y^(3)) = v`, so that, `-(3)/(y^(4))(dy)/(dx) = (dv)/(dx)` <br> Our equation reads, <br> `-(1)/(3)(dv)/(dx) - (2v)/(x) = 1` <br> `rArr (dv)/(dx) + (6)/(x) v = -3` <br> which is a linear differential equation in v. <br> `IF = e^(6int (dx)/(x)) = e^(6 ln x) = x^(6)` <br> Multiplying the equation by IF and integrating, we get <br> `vx^(6) = -3int x^(6) dx + k`, k being a constant of integration. <br> `= -3 (x^(7))/(7) + k` <br> `rArr (1)/(y^(3)) x^(6) = -(3)/(7) x^(7) + k` <br> `rArr (3)/(7)x^(7) + (x^(6))/(y^(3)) = k` <br> `:. x^(6)(3x+(7)/(y^(3))) = lambda, lambda` being another constant.