$\text { Find the integrating factor of the differential equation } ( x + 3 y ) d x - 3 \cot x d y = 0 \text { that }$ is a function of x or y alone.
A) $1 + 3 \cot x$
B) $\cos x$
C) $\sin x$
D) $- 3 \cos x$
E) $- 3 \sin x$

Question

$\text { Find the integrating factor of the differential equation } ( x + 3 y ) d x - 3 \cot x d y = 0 \text { that }$ is a function of x or y alone. 
A) $1 + 3 \cot x$
B) $\cos x$
C) $\sin x$
D) $- 3 \cos x$
E) $- 3 \sin x$

Quizplus · Accepted Answer

The given differential equation can be written in the form $M dx + N dy = 0$, where $M = x + 3y$ and $N = -3\cot x$. To find an integrating factor that depends only on $x$ or $y$, we check if $\frac{1}{N}(\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x})$ is a function of $x$ or $y$ alone. Here, $\frac{\partial M}{\partial y} = 3$ and $\frac{\partial N}{\partial x} = 3\csc^2 x$. Thus, $\frac{1}{-3\cot x}(3 - 3\csc^2 x)$ simplifies to a function of $x$ alone. The integrating factor that makes the equation exact can be found by integrating this expression with respect to $x$, leading to an integrating factor of $\sin x$, which is option C.

Find the integrating factor of the differential equation (x+3y)dx−3cot⁡xdy=0 that \text { Find the integrating factor of the differential equation } ( x + 3 y ) d x - 3 \cot x d y = 0 \text { that } Find the integrating factor of the differential equation (x+3y)dx−3cotxdy=0 that

$\text { Find the integrating factor of the differential equation } ( x + 3 y ) d x - 3 \cot x d y = 0 \text { that }$