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solve using first order homogenous equation

The differential equation presented in the image is:

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

We need to solve this using methods for first-order homogeneous differential equations. Let's work through this:

1. **Rewriting the Equation**: The equation is of the form:

\[
M(x, y) \, dx + N(x, y) \, dy = 0
\]

where:
- \( M(x, y) = \sin(2xy) \)
- \( N(x, y) = y^2 - x^2 \)

2. **Checking for Homogeneity**: A function is homogeneous if all terms in \( M(x, y) \) and \( N(x, y) \) are of the same degree. The term \( \sin(2xy) \) suggests that substitution methods involving homogeneity might be useful here. We typically introduce substitutions like \( v = \frac{y}{x} \) to simplify.

Let's solve this equation by making a substitution \( y = vx \) (or \( v = \frac{y}{x} \)) and find the general solution.from sympy import symbols, sin, Function, dsolve, Eq, Derivative

# Defining the variables
x, y, v = symbols('x y v')
f = Function('f')(x)

# Original Differential Equation Components
M = sin(2 * x * y)  # M(x, y)
N = y**2 - x**2     # N(x, y)

# Setting up the equation as dy/dx = -M/N for solving
dy_dx = -M/N

# Performing a substitution y = vx
# where y = v*x so that dy/dx = v + x*dv/dx

# Solving the homogeneous differential equation after substitution
dy_dx_expr = dy_dx.subs(y, v * x).simplify()
dy_dx_expr

Learn how to solve the first-order homogeneous differential equation sin(2xy) dx + (y^2 - x^2) dy = 0 using substitution methods and homogeneous equation techniques. This problem involves differential equations and is suitable for advanced high school or college students.

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