Question

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(1 point) In this problem we consider an equation in differential form M dx + N dy-0 (- (xy' +y)) dx + (- (x2y + x))dy 0 Find If the problem is exact find a function F(x, y) whose differential, d F(x, y) is the left hand side of the differential equation. That is, level curves F(x, y C, give implicit general solutions to the differential equation. If the equation is not exact, enter NE otherwise find F(x, y) (note you are not asked to enter C) F(x, y

Answer 1

(- (x y^2 + y)) dx + (-(x^2 y + x)) dy = 0 rightarrow (1) M dx + N dy = 0 rightarrow (2) comparing equation (1) and equation (2) we get M(x y) = -x y^2 - y N(x y) = -x^2 - x M_y (x y) = partial differential (x y)/partial differential y = partial differential/partial differential y (-x y^2 - y) M_y (x y) = -2 x y - 1 N_x (x y) = partial differential N (x y)/partial differential x = partial differential /partial differential x (-x^2 y - x) N_x (x y) = -2 x y - 1 here M_y (x y) = N_x (x y) here the partial differential equation is exact by exact differential equation F(x y) = integral m(x y) dx + integral N (x y) dy = c y as constant only those terms which do not contain x rightarrow F (x y) = integral (-x y^2 -y) dx -