MAT244--2019F > Quiz-2

TUT0502 Quiz2

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Xinqiao Li:
Determine whether the equation given below is exact. If it is exact, find the solution
$$(e^xsin(y)-2ysin(x))-(3x-e^xsin(y))y'=0\\ (e^xsin(y)-2ysin(x))dx-(3x-e^xsin(y))dy=0$$
Let $M(x,y) = e^xsin(y)-2ysin(x)$

Let $N(x,y) = -(3x-e^xsin(y)) = e^xsin(y)-3x$
$$M_y(x,y) = \frac{\partial}{\partial y} M(x,y)= \frac{\partial}{\partial y} (e^xsin(y)-2ysin(x)) = e^xcos(y)-2sin(x)\\ N_x(x,y) = \frac{\partial}{\partial x} N(x,y) = \frac{\partial}{\partial x} (e^xsin(y)-3x) = e^xsin(y)-3$$
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.