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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.

Eliminate Constants and write the Differential Equation: 1. xy = C(x-1)(y-1), 2. y = C1x^2 + C2e^(-x), 3. x = C1sin(y + C2)

Learn how to eliminate constants and derive differential equations from implicit equations like xy = C(x-1)(y-1), y = C1x^2 + C2e^(-x), and x = C1sin(y + C2). Step-by-step solutions using implicit differentiation and product rule. Suitable for university-level students.

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