Let y=y(x) be the solution of the differential equation x d y d x - 2 y=x³ (2-x³ ) ² y, x 0 . If y(2)=0 , then (y(1)) is equal to

Step-by-step solution

Divide the equation by ^2 y : x ^2 y dy dx - 2 y = x^3(2-x^3) . Let v = y , so ^2 y dy dx = dv dx . The equation becomes x dv dx - 2v = x^3(2-x^3) . Dividing by x : dv dx - 2v x = x^2(2-x^3) . Using integrating factor = x⁻² : d dx (x⁻²v) = 2 - x^3 . Integrating: x⁻²v = 2x - x^4 4 + C . From y(2) = 0 , we have 0 = 0 = 16 - 16 + 4C , giving C = 0 . Therefore y = 2x^3 - x^6 4 . At x = 1 : (y(1)) = 2 - 1 4 = 7 4 .