exact vs non-exact D.E.

Question

Classify the following equation as exact or not exact.

(9x^2 -18xy - 2x - 5y^2 +1) dy/dx + (-24x^2 + 18xy - 7y^2 + 4y + 4) = 0

From what i understand you take the partial m with respect to y for the left and partial n with respect to x for the right and compare. I keep getting the wrong answer, could someone please clarify? Thanks.

Answer 1

Let us put the given differential equation in the form M(x ,y) d x + N (x, y ) d y = 0

(-24x^2 + 18xy - 7y^2 + 4y + 4) d x + (9x^2 -18xy - 2x - 5y^2 +1) d y = 0

M(x ,y) = -24x^2 + 18xy - 7y^2 + 4y + 4

N (x, y ) = 9x^2 -18xy - 2x - 5y^2 +1

Then ∂ M /∂y = 18x -14 y + 4

and ∂ N /∂x = 18 x - 18 y - 2

Now notice that ∂ M /∂y ≠ ∂ N /∂x. Hence the given differential equation is not an exact one.