Question

Struggling with this differential equations problem. Can't find the integrating factor to continue

Solve the equation. (4x2 +2y+ 2y2dx + (x + 2xy)dy 0 An implicit solution in the form F(x,y) C is by multiplying by the integrating factor C, where C is an arbitrary constant, and (Type an expression using x and y as the variables.)

Answer 1

(4 x^2 + 2 y + 2y^2) dx + (x + 2xy)dy = 0 (1) m = 4 x^2 + 2y + 2y^2 , N = x + 2 xy partial differential M/ partial differential y = 2 + 4y partial differential/ partial differential x = 1 + 2y 1/N (partial differential M/ partial differential y - partial differential N/ partial differential x) = 1/x (1 + 2y) (2 + 4y - 1 - 2y) = 1/x (1 + 2y) (1 + 2y) = 1/x I F = e^integral 1/x dx = e^ln x = x multiply eq^n (1) by x rightarrow (4 x^3 + 2 x y + 2 x y^2) dx + (x^2 + 2 x^2 y) dy = 0 rightarrow d (x^4) + d (x^2 y) + d (x^2 y^2) = d (c) on integrating rightarrow x^4 + x^2 y + x^2 y^2 = c where c is integrating constant