# Differential equation with separable, probably wrong answer in book I have a differential equation: (dy)/(dx) = y log(y)cot(x)

Differential equation with separable, probably wrong answer in book
I have a differential equation:
$\frac{dy}{dx}=y\mathrm{log}\left(y\right)\mathrm{cot}\left(x\right)$
I'm trying solve that equation by separating variables and dividing by $y\mathrm{log}\left(y\right)$
$dy=y\mathrm{log}\left(y\right)\mathrm{cot}\left(x\right)dx$
$\frac{dy}{y\mathrm{log}\left(y\right)}=\mathrm{cot}\left(x\right)dx$
$\mathrm{cot}\left(x\right)-\frac{dy}{y\mathrm{log}\left(y\right)}=0$
Where of course y>0 regarding to division
Beacuse:
$\int \frac{dy}{y\mathrm{log}\left(y\right)}=\mathrm{ln}|\mathrm{ln}\left(y\right)|+C$
and:
$\int \mathrm{cot}\left(x\right)dx=\mathrm{ln}|\mathrm{sin}\left(x\right)|+C$
So:
$\mathrm{ln}|\mathrm{sin}\left(x\right)|-\mathrm{ln}|\mathrm{ln}\left(y\right)|=C$
$\mathrm{ln}|\frac{\mathrm{sin}\left(x\right)}{\mathrm{ln}\left(y\right)}|=C$
$\frac{\mathrm{sin}\left(x\right)}{\mathrm{ln}\left(y\right)}=±{e}^{C}$
$d=±{e}^{C}$
$\mathrm{sin}\left(x\right)=d\mathrm{ln}\left(y\right)$
$\frac{\mathrm{sin}\left(x\right)}{d}=\mathrm{ln}\left(y\right)$
${e}^{\frac{\mathrm{sin}\left(x\right)}{d}}=y$
This is my final answer. I have problem because in book from equation comes the answer to exercise is:
$y={e}^{c\mathrm{sin}\left(x\right)}$
Which one is correct?
I will be grateful for explaining Best regards
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

indivisast7
Note that
$\int \frac{du}{u}=\mathrm{ln}|u|+{c}_{1}=\mathrm{ln}|u|+\mathrm{ln}|c|=\mathrm{ln}|cu|.$
$\mathrm{cot}x\phantom{\rule{thinmathspace}{0ex}}dx=\frac{dy}{y\mathrm{ln}y}$
$\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}\frac{d\left(\mathrm{sin}x\right)}{\mathrm{sin}x}=\frac{d\left(\mathrm{ln}y\right)}{y\mathrm{ln}y}$
we integrate both sides:
$\int \frac{d\left(\mathrm{sin}x\right)}{\mathrm{sin}x}\phantom{\rule{thinmathspace}{0ex}}dx=\int \frac{d\left(\mathrm{ln}y\right)}{\mathrm{ln}y}\phantom{\rule{thinmathspace}{0ex}}dx$
$\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}\mathrm{ln}|a\mathrm{sin}x|=\mathrm{ln}|b\mathrm{ln}y|$
$\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}a\mathrm{sin}x=b\mathrm{ln}y,\phantom{\rule{1em}{0ex}}\left(1\right)$
$\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}\mathrm{ln}y=c\mathrm{sin}x$
$\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}y={e}^{c\mathrm{sin}x}$
where a,b,c are arbitrary constants. Note that (1) includes all possible solutions including the case a=0.