Final answer:
The given differential equation is exact because the partial derivative of the function multiplied by dt with respect to y and the partial derivative of the function multiplied by dy with respect to t are equal.
Step-by-step explanation:
To determine if the given differential equation is exact, we need to check if the partial derivatives of the functions multiplied by dt and dy are equal. The differential equation is (4t³y − 18t² − y)dt + (t⁴ + 3y² − t)dy = 0. For an equation of the form M(t, y)dt + N(t, y)dy = 0 to be exact, there must exist a function F(t, y) such that M = ∂F/∂t and N = ∂F/∂y. Let's check the partial derivatives of M with respect to y and N with respect to t.
Since ∂M/∂y = ∂N/∂t, the differential equation is indeed exact.