To determine which of these equations are separable, we need to check whether the equation can be rewritten in the form:

$\frac{dy}{dx} = g(x) h(y)$ or some similar variation for other variables like $t$ and $s$. Here's a step-by-step analysis of each equation:

1. Equation 1: $\frac{dy}{dx} = 2y^2 - 3y - 1$

   • This equation is separable because it can be written as:
     $\frac{dy}{2y^2 - 3y - 1} = dx$ which separates $y$ and $x$ on different sides.

2. Equation 2: $\frac{dy}{dx} - \sin(x^3 + y^4) = 0$

   • This equation is not separable because the terms involving $x$ and $y$ are mixed within the sine function, making it impossible to separate them.

3. Equation 3: $\frac{dx}{dt} = \frac{7te^{t + x}}{t^4 - 3}$

   • This equation is not separable because $x$ and $t$ are both present in the exponential function, preventing separation.

4. Equation 4: $\frac{ds}{dt} = 4t\ln(s^3 t) + 5t^2$

   • This equation is not separable due to the $\ln(s^3t)$ term, which mixes $t$ and $s$.

5. Equation 5: $z^2 + \frac{dz}{dr} = z + \frac{1}{zr}$

   • This equation is not separable due to the presence of both $z$ and $r$ on the right-hand side in a non-separable form.

6. Equation 6: $(xy^2 + 3y^2)dy - \frac{2x}{dx} = 0$

   • This equation can be rearranged as: $\frac{dy}{dx} = \frac{2x}{xy^2 + 3y^2}$ which is separable, as it can be written as: $\frac{(xy^2 + 3y^2)dy}{2x} = dx$ allowing separation of $x$ and $y$.

Separable Equations:

• Equation 1: $\frac{dy}{dx} = 2y^2 - 3y - 1$
• Equation 6: $(xy^2 + 3y^2)dy - \frac{2x}{dx} = 0$

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Would you like a detailed explanation for one of these separations? Here are some related questions:

1. How do you solve separable differential equations?
2. What are common techniques for non-separable differential equations?
3. Can every differential equation be made separable under certain transformations?
4. What role do initial conditions play in solving separable equations?
5. How does separation of variables apply in partial differential equations?

Tip: Always check if terms involving different variables can be separated to opposite sides of the equation, this will quickly tell you if the equation is separable.