:
### Step 1: Rearranging the Equation
We start with the given equation:
\[
y \, dx - x \, dy + (1 + x^2) \, dx + x^2 \sin y \, dy = 0
\]
We can group the terms involving \( dx \) and \( dy \):
\[
(y + 1 + x^2) \, dx + (-x + x^2 \sin y) \, dy = 0
\]
### Step 2: Dividing by \( x^2 \)
To simplify the equation, we can divide the entire equation by \( x^2 \):
\[
\frac{y}{x^2} \, dx + \left(\frac{1}{x^2} + 1\right) \, dx + \left(-\frac{x}{x^2} + \sin y\right) \, dy = 0
\]
This simplifies to:
\[
\frac{y}{x^2} \, dx + \left(\frac{1 + x^2}{x^2}\right) \, dx + \left(-\frac{1}{x} + \sin y\right) \, dy = 0
\]
### Step 3: Factoring Out Common Terms
Now, we can factor out the common terms:
\[
-\left(x \, dy - y \, dx\right) + \left(1 + \frac{1}{x^2}\right) \, dx + \sin y \, dy = 0
\]
### Step 4: Recognizing the Derivative
Notice that the expression \( x \, dy - y \, dx \) is the differential of \( \frac{y}{x} \):
\[
\frac{d}{dx}\left(\frac{y}{x}\right) = \frac{x \, dy - y \, dx}{x^2}
\]
Thus, we can rewrite the equation:
\[
-\frac{d}{dx}\left(\frac{y}{x}\right) + \left(1 + \frac{1}{x^2}\right) \, dx + \sin y \, dy = 0
\]
### Step 5: Integrating the Equation
Now we can integrate both sides:
\[
-\frac{y}{x} + \int \left(1 + \frac{1}{x^2}\right) \, dx + \int \sin y \, dy = C
\]
Calculating the integrals:
\[
-\frac{y}{x} + x - \frac{1}{x} + \cos y = C
\]
### Step 6: Rearranging the Final Solution
Rearranging gives us:
\[
\frac{y}{x} + \frac{1}{x} - x + \cos y = C
\]
### Final Solution
Thus, the solution to the differential equation is:
\[
\frac{y}{x} + \frac{1}{x} - x + \cos y = C
\]

Solve: `ydx-xdy+(1+x^2)dx+x^2sinydy=0`

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The solution of ydx-xdy+(1+x^(2))dx+x^(2)siny dy=0, is given by

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