(Solved): Determine whether the given differential equation is exact. If it is exact, solve it. 8xex -y + 6x ...

Determine whether the given differential equation is exact. If it is exact, solve it. 8xex -y + 6x² X Step 1 Recall that a differential equation of the form M(x, y) dx + N(x, y) dy = 0 is exact if there is a region R of the xy-plane af af such that there exists a function f(x, y) defined on R such that ?x dx By Theorem 2.4.1, the existence of the function f is equivalent to the conditions that M and N are continuous on a rectangular region a < x < b, c < y < d and the following partial derivatives are equal. x dy + y - M(x, y) = ?M ?N ?y ?x We are given a differential equation and must rewrite it in the form M(x, y) dx + N(x, y) dy = 0. dy dx x dy = ( Find the functions M and N. N(x, y) = = X- ()xex - 6x²) dx = 8xe* - 6x² = M(x, y) and = N(x, y). ?y 0 8xex -y + 6x² (8xex - y + 6x²) dx