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$x\sin(y) \, dx + ((x^2) + 1) \cos (y) \, dy = 0$

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Add and subtract polynomials

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Q.

x\sin(y) \, dx + ((x^2) + 1) \cos (y) \, dy = 0

Separate variables: We are given a differential equation $x\sin(y) \, dx + ((x^2) + 1) \cos(y) \, dy = 0$ . To solve this, we will try to separate the variables $x$ and $y$ so that we can integrate both sides with respect to their own variables.
Divide by $x\sin(y) \cos(y)$ : First, we divide the entire equation by $x\sin(y) \cos(y)$ to separate the variables. This gives us: $\newline$ $\frac{dx}{x} + \frac{(x^2) + 1}{x\sin(y)} dy = 0$ .
Correct separation: We notice that the term $\frac{(x^2) + 1}{x\sin(y)}$ is not properly separated because it contains both $x$ and $y$ . We need to correct this by dividing the equation by $\cos(y)$ and multiplying by $\sin(y)$ to get: $\frac{\sin(y)}{\cos(y)} \frac{dx}{x} + \left(\frac{\sin(y)}{x}\right)((x^2) + 1) dy = 0$ .

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