Question

3: Problem 4 Previous Problem List Next (1 point) Consider the implicit differential equation (15y +56xy)dx (36ry 64x2 )dy0 Show that xpy! is an integrating factor of this equation where p = and q = Now multiply the equation by the integrating factor xy that you have found and then integrate the resulting equation to get a solution in implicit form. where C is a constant of integration. Note that credit is only given if your third answer is obtained as instructed above. Do not renormalize your answer

Answer 1

(15y^3 + 56xy)dx +(36xy^2 + 6 + x^2)dy = 0 x^p y^q is an integrating factor rightarrow x^p y^q (15y^3 + 56xy)dx + x^py^q(36xy^2 + 64x^2)dy = 0 is exact m = 15 x^py^q + 3 + 56x^p + 1y^q + 1 N = 36x^p + 1 y^q + 2 + 64x^p + 2y^q my = nx 15(q + 3)x^py^q + 2 + 56(q + 1)x^p + 1y^q = 36(p + 1)x^py^q + 2 + 6(p + 2x^p + 1 y^q comparing 15(q + 3) = 36(p + 1) 56(q + 1) = 64(p + 2) rightarrow 5q + 15 = 12p + 12 rightarrow 12p-5q = 3 rightarrow 7q + 7 = 8p + 16 rightarrow 2p-7q = -9 after solving we get p = 3/2, q = 3 IF is x^3/2 y^3 sol^n is integral mdx + integral ndy = c y = constant rightarrow integral(15x^3/2y^6 + 56x^5/2y^4)dx + integral dy = c rightarrow x^5/2/5/2 y^6 + 56 x^7/2/7/2 y^4 = c rightarrow 6x^5/2y^4 + 16x^7/2y^4 = c