Differential equation with separable, probably wrong answer in book

I have a differential equation:

$$\frac{dy}{dx}=y\mathrm{log}(y)\mathrm{cot}(x)$$

I'm trying solve that equation by separating variables and dividing by $y\mathrm{log}(y)$

$$dy=y\mathrm{log}(y)\mathrm{cot}(x)dx$$

$$\frac{dy}{y\mathrm{log}(y)}=\mathrm{cot}(x)dx$$

$$\mathrm{cot}(x)-\frac{dy}{y\mathrm{log}(y)}=0$$

Where of course y>0 regarding to division

Beacuse:

$$\int \frac{dy}{y\mathrm{log}(y)}=\mathrm{ln}|\mathrm{ln}(y)|+C$$

and:

$$\int \mathrm{cot}(x)dx=\mathrm{ln}|\mathrm{sin}(x)|+C$$

So:

$$\mathrm{ln}|\mathrm{sin}(x)|-\mathrm{ln}|\mathrm{ln}(y)|=C$$

$$\mathrm{ln}|\frac{\mathrm{sin}(x)}{\mathrm{ln}(y)}|=C$$

$$\frac{\mathrm{sin}(x)}{\mathrm{ln}(y)}=\pm {e}^{C}$$

$d=\pm {e}^{C}$

$$\mathrm{sin}(x)=d\mathrm{ln}(y)$$

$$\frac{\mathrm{sin}(x)}{d}=\mathrm{ln}(y)$$

$${e}^{\frac{\mathrm{sin}(x)}{d}}=y$$

This is my final answer. I have problem because in book from equation comes the answer to exercise is:

$$y={e}^{c\mathrm{sin}(x)}$$

Which one is correct?

I will be grateful for explaining Best regards

I have a differential equation:

$$\frac{dy}{dx}=y\mathrm{log}(y)\mathrm{cot}(x)$$

I'm trying solve that equation by separating variables and dividing by $y\mathrm{log}(y)$

$$dy=y\mathrm{log}(y)\mathrm{cot}(x)dx$$

$$\frac{dy}{y\mathrm{log}(y)}=\mathrm{cot}(x)dx$$

$$\mathrm{cot}(x)-\frac{dy}{y\mathrm{log}(y)}=0$$

Where of course y>0 regarding to division

Beacuse:

$$\int \frac{dy}{y\mathrm{log}(y)}=\mathrm{ln}|\mathrm{ln}(y)|+C$$

and:

$$\int \mathrm{cot}(x)dx=\mathrm{ln}|\mathrm{sin}(x)|+C$$

So:

$$\mathrm{ln}|\mathrm{sin}(x)|-\mathrm{ln}|\mathrm{ln}(y)|=C$$

$$\mathrm{ln}|\frac{\mathrm{sin}(x)}{\mathrm{ln}(y)}|=C$$

$$\frac{\mathrm{sin}(x)}{\mathrm{ln}(y)}=\pm {e}^{C}$$

$d=\pm {e}^{C}$

$$\mathrm{sin}(x)=d\mathrm{ln}(y)$$

$$\frac{\mathrm{sin}(x)}{d}=\mathrm{ln}(y)$$

$${e}^{\frac{\mathrm{sin}(x)}{d}}=y$$

This is my final answer. I have problem because in book from equation comes the answer to exercise is:

$$y={e}^{c\mathrm{sin}(x)}$$

Which one is correct?

I will be grateful for explaining Best regards