Let the solution curve y = y x of the differential equation 4 + x 2 d y - 2 x x 2 + 3 y + 4 d x = 0 pass through the origin. Then y 2 is equal to _____.

Step-by-step solution

Given 4 + x 2 d y - 2 x x 2 + 3 y + 4 d x = 0 x 2 + 4 d y d x = 2 x 3 + 6 x y + 8 x x 2 + 4 d y d x - 6 x y = 2 x 3 + 8 x d y d x - 6 x x 2 + 4 y = 2 x 3 + 8 x x 2 + 4 This is of the form of linear differential equation I.F. = e - ∫ 6 x x 2 + 4 d x = e - 3 log e x 2 + 4 = e log e x 2 + 4 - 3 = 1 x 2 + 4 3 So the solution of the differential equation will be y . 1 x 2 + 4 3 = ∫ 2 x 3 + 8 x x 2 + 4 3 x 2 + 4 d x y x 2 + 4 3 = ∫ 2 x x 2 + 4 x 2 + 4 3 x 2 + 4 d x Let x 2 + 4 = t , 2 x d x = d t So y x