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To find the solution of the given differential equation:

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

We can rewrite the equation in the form \(Mdx + Ndy = 0\), where \(M\) and \(N\) are functions of \(x\) and \(y\):

\[ M = y \sin(2x) \]

\[ N = -(1 + y^2 + \cos^2x) \]

Now, we can check for the exactness of the equation by ensuring that \(\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}\):

\[ \frac{\partial M}{\partial y} = \sin(2x) \]

\[ \frac{\partial N}{\partial x} = 2y\sin(2x) - 2\cos(x)\sin(x) \]

Since \(\frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x}\), the equation is not exact. We can try to find an integrating factor (\(\mu\)) to make it exact.

\[ \mu = e^{\int \left(\frac{\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}}{N}\right)dy} \]

\[ \mu = e^{\int \frac{\sin(2x) - (y\sin(2x) - \cos(x)\sin(x))}{-(1 + y^2 + \cos^2x)} \, dy} \]

After finding the integrating factor, you can multiply the entire equation by it and then attempt to solve for \(y\). Note that the integration process can be complex, and the solution may not be straightforward. If you have specific values for \(x\) and \(y\) or initial conditions, it might help in obtaining a particular solution.

Step-by-step explanation:

Certainly! Let's solve the given differential equation step by step:

\[y\sin(2x)dx - (1 + y^2 + \cos^2 x)dy = 0\]

1. Rearrange the terms:

\[y\sin(2x)dx = (1 + y^2 + \cos^2 x)dy\]

2. Separate variables:

\[\frac{1}{\sin(2x)}dx = \frac{1+y^2+\cos^2x}{y}dy\]

3. Integrate both sides with respect to x and y:

\[\int \frac{1}{\sin(2x)} \,dx = \int \frac{1+y^2+\cos^2x}{y} \,dy\]

The integrals on both sides will give the solution. Note that the integration might involve trigonometric identities and substitution.

4. Evaluate the integrals:

Solve the integrals on both sides. This may involve using trigonometric identities or substitution techniques.

5. Write down the final solution:

Express the result in terms of y and x, incorporating any constants of integration that may arise during the process.

Without specific values or further information about the integrals, it's challenging to provide an exact solution.