Toronto Math Forum
MAT2442019F => MAT244Test & Quizzes => Quiz2 => Topic started by: Xinqiao Li on October 04, 2019, 02:00:01 PM

Determine whether the equation given below is exact. If it is exact, find the solution
$$
(e^xsin(y)2ysin(x))(3xe^xsin(y))y'=0\\
(e^xsin(y)2ysin(x))dx(3xe^xsin(y))dy=0
$$
Let $M(x,y) = e^xsin(y)2ysin(x)$
Let $N(x,y) = (3xe^xsin(y)) = e^xsin(y)3x$
$$
M_y(x,y) = \frac{\partial}{\partial y} M(x,y)= \frac{\partial}{\partial y} (e^xsin(y)2ysin(x)) = e^xcos(y)2sin(x)\\
N_x(x,y) = \frac{\partial}{\partial x} N(x,y) = \frac{\partial}{\partial x} (e^xsin(y)3x) = e^xsin(y)3
$$
Clearly, we see that $ e^xcos(y)2sin(x) \neq e^xsin(y)3$
Therefore, $M_y(x,y) \neq N_x(x,y)$
By definition of exact, we can conclude the equation is not exact.