Question

(2) (13 points) Determine whether the given differential equation s e If it is exact, solve it:

Answer 1

(X^3 + y^3) dx + 3 xy^2 dy = 0 M dx + N dy = 0 Where M = X^3 + y^3, N = 3xy^2 partial differential M/ partial differential y = 3y^2 partial differential N/ partial differential x = 3y^2 Therefore partial differential M/ partial differential y = partial differential N/ partial differential X = > The equation is exact Let Psi (x, y) = integral N dy = 3 times y^2 dy = 3 times integral y^2 dy = 3 times y^3/3 + h(x) = xy^3 + h(x) partial differential psi/dx = y^3 + h prime(x) = M = x^3 + y^3 = > h prime (x) = x^3 = > h(x) = integral x^3 = x^4/4 + c_1 c_1 constant Therefore Psi (x, y) = x^4/4 + xy^3 + c_1 Therefore solution to the differential equation is Psi (x, y) = k k constant = > X^4/4 + Xy^3 + C_1 = k let C = k - C_1