We are given the following differential equation:

$\frac{dy}{dx} = \frac{xy + 3x - y - 3}{xy - 2x + 4y - 8}$

Let's break this down step by step.

1. Simplify the equation: Check if the numerator and denominator can be simplified or factored. The goal is to see if we can separate the variables $x$ and $y$.

$\frac{dy}{dx} = \frac{xy + 3x - y - 3}{xy - 2x + 4y - 8}$

Looking closely at the numerator and denominator, they don't appear to simplify easily. So we'll look into the possibility of solving it by a substitution method or separating variables if possible.

2. Attempt variable substitution: Often for such equations, trying a substitution like $u = \frac{y}{x}$ (to convert the equation into a simpler form) is helpful. Let's try that:

Let $u = \frac{y}{x}$, then:

$y = ux \quad \text{and} \quad \frac{dy}{dx} = u + x\frac{du}{dx}$

Substitute $y = ux$ into the original equation:

$u + x\frac{du}{dx} = \frac{x(ux) + 3x - (ux) - 3}{x(ux) - 2x + 4(ux) - 8}$

Simplify both numerator and denominator:

$u + x\frac{du}{dx} = \frac{x^2 u + 3x - ux - 3}{x^2 u - 2x + 4ux - 8}$

At this point, we can try simplifying further, but the equation doesn't seem to easily separate the variables.

Conclusion:

This differential equation is not easily separable or solvable by standard elementary methods without resorting to more advanced techniques or numerical methods. Solving this analytically might require more advanced methods, potentially using computational tools for exact solutions.

Would you like me to attempt solving this numerically or using another method?

Related Questions:

1. How can we solve differential equations numerically (e.g., using Euler's method)?
2. What are common substitutions for solving first-order differential equations?
3. What is the method of integrating factors in solving linear differential equations?
4. How do we recognize when a differential equation can be separated?
5. How can partial fractions help simplify certain rational differential equations?

Tip:

Substitutions like $u = \frac{y}{x}$ or $y = vx$ can sometimes transform complex differential equations into simpler forms for easier solving.