You're correct in thinking that this might be a case for a change of variables, particularly a substitution that could simplify the equation and make it homogeneous. Let's solve the differential equation step by step:

Given equation:

$\frac{dy}{dx} = \frac{-2y + x}{2x + y + 1}$

Step 1: Substitution

We try a substitution that can simplify the equation, often one that expresses the relationship between $y$ and $x$ in terms of their ratio. Let:

$v = \frac{y}{x}$

This implies:

$y = vx$

Now differentiate both sides with respect to $x$ using the product rule:

$\frac{dy}{dx} = v + x \frac{dv}{dx}$

Step 2: Substitute into the equation

Now, we substitute $y = vx$ and $\frac{dy}{dx} = v + x \frac{dv}{dx}$ into the original differential equation:

$v + x \frac{dv}{dx} = \frac{-2(vx) + x}{2x + (vx) + 1}$

Simplifying both the numerator and denominator:

• Numerator: $-2(vx) + x = x(-2v + 1)$
• Denominator: $2x + vx + 1 = x(2 + v) + 1$

Thus, the equation becomes:

$v + x \frac{dv}{dx} = \frac{x(-2v + 1)}{x(2 + v) + 1}$

Step 3: Simplify the equation

Now, cancel $x$ from both the numerator and denominator:

$v + x \frac{dv}{dx} = \frac{-2v + 1}{2 + v + \frac{1}{x}}$

At this point, this equation is a bit tricky to solve analytically by hand without more advanced techniques (and depending on how you want to approach solving it), but it does suggest a path forward where $v = \frac{y}{x}$ is the key to simplifying this expression. You can then proceed with methods of separation of variables, or even numerical methods, if necessary.

Would you like more details about further simplification or solving for $v$ explicitly?