Exact equation with trig function

[find_the_fun]

Determine whether the given differential equation is exact and if so solve it.

\(\displaystyle (\tan{x}-\sin{x}\sin{y}) dx+\cos{x}\cos{y} dy=0\)

I got \(\displaystyle \cos{x}\sin{y}+\sec^2{x}=c\) but the answer key has \(\displaystyle -ln|\cos{x}|+\cos{x}\sin{y}=c\) where did \(\displaystyle ln\) come from?

 

[Ackbach]

Determine whether the given differential equation is exact and if so solve it.

\(\displaystyle (\tan(x)-\sin(x)\sin(y)) \, dx+\cos(x)\cos(y) \, dy=0\)

I got \(\displaystyle \cos(x)\sin(y)+\sec^2(x)=c\) but the answer key has \(\displaystyle -\ln|\cos(x)|+\cos(x)\sin(y)=c.\) Where did the \(\displaystyle \ln\) come from?

So we start out with
$$f_x(x,y)=\tan(x)-\sin(x)\sin(y).$$
Integrating w.r.t. $x$ yields $-\ln|\cos(x)|+\cos(x)\sin(y)$. Then you do the usual differentiation w.r.t. $y$, compare with $\cos(x) \cos(y)$, etc. That's where the logarithm came from.