Which of the following is an exact differential equation?

A) $( y \sin x - 3 \cot y ) d x + ( 3 x \sin y - \cos x ) d y$
B) $( y \sin x - 3 \cos y ) d x + ( 3 x \sin y - \cot x ) d y$
C) $( y \sin x - 3 \cos y ) d x + ( 3 x \sec y - \cos x ) d y$
D) $( y \sin x - 3 \cos y ) d x + ( 3 x \sin y - \cos x ) d y$
E) $( y \sec x - 3 \cos y ) d x + ( 3 x \sin y - \cos x ) d y$

Question

Which of the following is an exact differential equation?

A) $( y \sin x - 3 \cot y ) d x + ( 3 x \sin y - \cos x ) d y$
B) $( y \sin x - 3 \cos y ) d x + ( 3 x \sin y - \cot x ) d y$
C) $( y \sin x - 3 \cos y ) d x + ( 3 x \sec y - \cos x ) d y$
D) $( y \sin x - 3 \cos y ) d x + ( 3 x \sin y - \cos x ) d y$
E) $( y \sec x - 3 \cos y ) d x + ( 3 x \sin y - \cos x ) d y$

Quizplus · Accepted Answer

For an equation $ M(x, y) dx + N(x, y) dy = 0 $ to be exact, $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$. In choice D, $\frac{\partial}{\partial y}(y \sin x - 3 \cos y) = -3 \sin y$ and $\frac{\partial}{\partial x}(3 x \sin y - \cos x) = -\sin x$, which are equal, making it exact.

Which of the Following Is an Exact Differential Equation? (ysin⁡x−3cot⁡y)dx+(3xsin⁡y−cos⁡x)dy( y \sin x - 3 \cot y ) d x + ( 3 x \sin y - \cos x ) d y(ysinx−3coty)dx+(3xsiny−cosx)dy

Which of the Following Is an Exact Differential Equation?
$( y \sin x - 3 \cot y ) d x + ( 3 x \sin y - \cos x ) d y$