Let's go through the steps to eliminate constants and write the differential equations for each of the given equations.

1. First Equation:

$xy = C(x-1)(y-1)$

To eliminate the constant $C$, let's differentiate the equation implicitly with respect to $x$. Applying the product rule:

1. Differentiate both sides: $\frac{d}{dx}(xy) = \frac{d}{dx}\left(C(x-1)(y-1)\right)$

   Left-hand side: $\frac{d}{dx}(xy) = x\frac{dy}{dx} + y$

   Right-hand side (using product rule): $\frac{d}{dx}\left(C(x-1)(y-1)\right) = C\left((x-1)\frac{dy}{dx} + (y-1)\right)$

2. Equating both sides: $x\frac{dy}{dx} + y = C\left((x-1)\frac{dy}{dx} + (y-1)\right)$

Now, you can solve for $\frac{dy}{dx}$ to get the required differential equation.

2. Second Equation:

$y = C_1x^2 + C_2e^{-x}$

Here, $C_1$ and $C_2$ are constants. Let's differentiate both sides with respect to $x$:

1. Differentiate the left-hand side: $\frac{dy}{dx} = \frac{d}{dx}(C_1x^2 + C_2e^{-x})$

2. Right-hand side: $\frac{dy}{dx} = 2C_1x - C_2e^{-x}$

Thus, the differential equation is: $\frac{dy}{dx} = 2C_1x - C_2e^{-x}$

By eliminating the constants $C_1$ and $C_2$, you can express this as a second-order differential equation by differentiating again.

3. Third Equation:

$x = C_1 \sin(y + C_2)$

1. Differentiate both sides with respect to $y$: $\frac{dx}{dy} = C_1 \cos(y + C_2)$

Since $C_1$ is a constant, you can differentiate once more if needed to eliminate it, but for now, the first derivative should suffice.

Let me know if you would like to see any further steps or clarifications. Now, here are 5 related questions for practice:

1. How can we convert implicit derivatives into explicit form for certain differential equations?
2. What is the significance of higher-order derivatives in eliminating constants?
3. How can you determine the order of a differential equation from a given implicit form?
4. What types of functions commonly arise in solving second-order differential equations?
5. How would the procedure change if the equation contained multiple variables?

Tip: When eliminating constants, always check if you can differentiate implicitly with respect to one variable to reduce the equation step by step.