32. $cos(x)dx+(1 + \frac{2}{y})sin(x)dy = 0$

Explanation:

Step1: Separate variables

We rewrite the given differential equation $\cos(x)dx+(1 + \frac{2}{y})\sin(x)dy = 0$ as $\frac{\cos(x)}{\sin(x)}dx=-(1+\frac{2}{y})dy$.

Step2: Integrate both sides

Integrate $\int\frac{\cos(x)}{\sin(x)}dx$ and $\int-(1+\frac{2}{y})dy$. For $\int\frac{\cos(x)}{\sin(x)}dx$, let $u = \sin(x)$, then $du=\cos(x)dx$ and $\int\frac{\cos(x)}{\sin(x)}dx=\int\frac{du}{u}=\ln|u|=\ln|\sin(x)|+C_1$. For $\int-(1+\frac{2}{y})dy=-\int dy - 2\int\frac{1}{y}dy=-y - 2\ln|y|+C_2$. So we have $\ln|\sin(x)|=-y - 2\ln|y|+C$.

Step3: Simplify the result

We can rewrite it as $\ln|\sin(x)|+y + 2\ln|y|=C$, or $\ln|\sin(x)y^{2}|+y = C$.

Answer:

$\ln|\sin(x)y^{2}|+y = C$