(xdx+sin²y/x)(ydx-xdy)=0 solve this equation using homogeneous coefficient, differential equations

To solve the differential equation 

\[
\left( x \, dx + \frac{\sin^2 y}{x} \right) \left( y \, dx - x \, dy \right) = 0,
\]

we start by expanding the expression. The equation is a product of two terms, and for the product to be zero, at least one of the terms must be zero. Therefore, we have two cases to consider:

1. \( x \, dx + \frac{\sin^2 y}{x} = 0 \)
2. \( y \, dx - x \, dy = 0 \)

Let's solve each case separately.

### Case 1: \( x \, dx + \frac{\sin^2 y}{x} = 0 \)

Rewriting this differential equation gives:

\[
x \, dx = -\frac{\sin^2 y}{x}.
\]

Separating the variables:

\[
x^2 \, dx = -\sin^2 y \, dy.
\]

To separate variables, integrate both sides:

\[
\int x^2 \, dx = -\int \sin^2 y \, dy.
\]

Using the integrals:

\[
\frac{x^3}{3} = -\int \sin^2 y \, dy.
\]

Recall that:

\[
\int \sin^2 y \, dy = \frac{y}{2} - \frac{\sin(2y)}{4} + C.
\]

Thus:

\[
\frac{x^3}{3} = - \left( \frac{y}{2} - \frac{\sin(2y)}{4} \right) + C_1.
\]

This gives a general solution for Case 1.

### Case 2: \( y \, dx - x \, dy = 0 \)

This is a homogeneous differential equation. We can rewrite it as:

\[
\frac{dy}{dx} = \frac{y}{x}.
\]

Separating variables:

\[
\frac{dy}{y} = \frac{dx}{x}.
\]

Integrating both sides:

\[
\ln |y| = \ln |x| + C_2.
\]

Exponentiating both sides:

\[
|y| = e^{C_2} |x|.
\]

Let \( C = e^{C_2} \). Then:

\[
y = Cx.
\]

This gives another family of solutions, where \( y \) is proportional to \( x \).

### General Solution

The general solution to the differential equation is the combination of the two cases:

1. \(\frac{x^3}{3} = - \left( \frac{y}{2} - \frac{\sin(2y)}{4} \right) + C_1\)
2. \(y = Cx\)

These solutions represent the complete set of solutions for the differential equation.

Would you like more details on any of these steps or have any questions?

### 5 Related Questions

1. How do you determine if a differential equation is homogeneous?
2. What is the method for integrating trigonometric functions?
3. How do you find particular solutions for differential equations given initial conditions?
4. What are the properties of linear versus non-linear differential equations?
5. Can you solve the given differential equation using another method, such as separation of variables or an integrating factor?

### Tip

When solving differential equations, always check if you can simplify or separate the variables first. Homogeneous and separable forms often allow for straightforward integration and finding solutions.

Explore how to solve the differential equation (xdx + sin²y/x)(ydx - xdy) = 0 using the homogeneous coefficient method. This solution involves detailed steps including separating variables and integrating both sides. Suitable for advanced undergraduate students and those studying differential equations.

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