Identify each of the differential equations as type (for example, separable, linear first order, linear second order, etc.), and then solve it. $$\sin ^{2} x d y+\left[\sin ^{2} x+(x+y) \sin 2 x\right] d x=0$$

Linear first-order differential equations have the general form \( \frac{dy}{dx} + P(x)y = Q(x) \). In these equations, the first-order derivative \( \frac{dy}{dx} \) is linear, which means \( y \) and its derivatives are raised only to the first power. The coefficients \( P(x) \) and \( Q(x) \) are functions of \( x \) only.

For example, consider a differential equation that can be written as \( \frac{dy}{dx} + P(x)y = Q(x) \). To solve this, one can use an integrating factor \( \rho(x) = e^{\bar{P(x) dx}} \), which converts the original equation into a solvable form. After multiplying the entire equation by the integrating factor, it can be integrated directly on both sides.

This concept is fundamental because it allows many real-world problems to be modeled and solved using linear equations.

Non-linear differential equations are equations where the unknown function and its derivatives appear non-linearly. This means that the function and its derivatives can be squared, cubed, or raised to some other power. They are generally more difficult to solve than linear equations. For instance, an equation of the form \( \frac{dy}{dx} = y^2 + x \) is non-linear.

Non-linear differential equations are prevalent in modeling complex systems like weather patterns, population dynamics, and fluid dynamics. These types of equations often require special methods of solution, which can include approximate, numerical, or graphical methods. Transformations and substitutions can sometimes simplify them into linear form or use specific techniques like separation of variables if they adhere to certain properties.

Separation of variables is a method to solve differential equations. It involves rearranging the equation so that each variable and its differential is on opposite sides of the equation. For the differential equation \( \frac{dy}{dx} = g(x)h(y) \), it can be rewritten as \( \frac{1}{h(y)} dy = g(x) dx \).

Once separated, each side can be integrated independently:

• Integrate \( \frac{1}{h(y)} dy \) with respect to \( y \)
• Integrate \( g(x) dx \) with respect to \( x \)

After integration, combine the results, applying constants of integration where necessary. Identifying when to use separation of variables simplifies otherwise complicated calculations and is powerful for problems where this method is applicable.

Integration is a fundamental operation in calculus used to find functions such as antiderivatives. In the context of differential equations, it’s a crucial step in solving both linear and non-linear equations. Once the differential equation is simplified appropriately using methods like separation of variables or integrating factors, integrating both sides can lead to the solution.

In practice, integration involves finding a function whose derivative matches the integrand. Common techniques include:

• Basic antiderivatives (e.g., \( \frac{1}{x} \) integrals)
• Integration by parts
• Partial fraction decomposition
• Using substitution (like \( u \)-substitution)

Mastering these techniques is essential because finding the correct antiderivative is often the key to solving the differential equation. Integration constants are crucial as they represent the general solution’s family of curves.