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Let's go through each equation and find the differential equation by eliminating arbitrary constants. We will then classify each resulting DE by its order, degree, and whether it is linear or nonlinear.

### 3. \( y = A x^2 + A \)
- **Step 1: Differentiate** with respect to \( x \) to eliminate the constant \( A \).
\[
\frac{dy}{dx} = 2Ax
\]
- **Step 2: Differentiate again** to eliminate \( A \):
\[
\frac{d^2y}{dx^2} = 2A
\]
Since \( A \) is now eliminated, we express \( A \) as \( \frac{1}{2} \frac{d^2y}{dx^2} \).

- **Resulting DE**: Substituting \( A \) into the original equation:
\[
y = \frac{1}{2} \frac{d^2y}{dx^2} x^2 + \frac{1}{2} \frac{d^2y}{dx^2}
\]

- **Order**: 2 (since the highest derivative is \( \frac{d^2y}{dx^2} \))
- **Degree**: 1 (no powers of derivatives)
- **Linearity**: Linear (no products or powers of the dependent variable \( y \) and its derivatives)

---

### 4. \( y = c_1 x + c_2 x \ln x \)
- **Step 1: Differentiate**:
\[
\frac{dy}{dx} = c_1 + c_2 (\ln x + 1)
\]
- **Step 2: Differentiate again**:
\[
\frac{d^2y}{dx^2} = \frac{c_2}{x}
\]
Now, eliminate \( c_1 \) and \( c_2 \) from the system.

- **Resulting DE**: Multiplying both sides by \( x \):
\[
x \frac{d^2y}{dx^2} = c_2
\]

Now substitute this into the second derivative expression:
\[
x \frac{d^2y}{dx^2} = \frac{dy}{dx} - \ln x
\]
This gives the differential equation:
\[
x \frac{d^2y}{dx^2} = \frac{dy}{dx} - \ln x
\]

- **Order**: 2
- **Degree**: 1
- **Linearity**: Linear

---

### 5. \( y = A \sin 3x + B \cos 3x \)
- **Step 1: Differentiate**:
\[
\frac{dy}{dx} = 3A \cos 3x - 3B \sin 3x
\]
- **Step 2: Differentiate again**:
\[
\frac{d^2y}{dx^2} = -9A \sin 3x - 9B \cos 3x
\]
Thus, we have:
\[
\frac{d^2y}{dx^2} = -9y
\]

- **Resulting DE**: 
\[
\frac{d^2y}{dx^2} + 9y = 0
\]

- **Order**: 2
- **Degree**: 1
- **Linearity**: Linear

---

### 6. \( y = c_1 e^{-3x} + c_2 e^{4x} \)
- **Step 1: Differentiate**:
\[
\frac{dy}{dx} = -3c_1 e^{-3x} + 4c_2 e^{4x}
\]
- **Step 2: Differentiate again**:
\[
\frac{d^2y}{dx^2} = 9c_1 e^{-3x} + 16c_2 e^{4x}
\]
Thus, we can express the second derivative as:
\[
\frac{d^2y}{dx^2} = 9y + 7\frac{dy}{dx}
\]

- **Resulting DE**:
\[
\frac{d^2y}{dx^2} - 7\frac{dy}{dx} = 9y
\]

- **Order**: 2
- **Degree**: 1
- **Linearity**: Linear

---

Would you like further details on any of these steps?

### Follow-up questions:
1. How does the classification of differential equations work in more complex cases?
2. Can you show how to solve one of these differential equations?
3. What is the general approach to finding a DE from a given function?
4. How can we solve second-order linear differential equations analytically?
5. How do we handle non-linear differential equations?

### Tip:
For second-order linear differential equations, knowing whether the DE has constant or variable coefficients can significantly affect the solving method.

Find the Differential Equation using elimination of arbitrary constants. Classify the resulting DE as to its order, degree, and linearity for the following equations: 
3. y = Ax^2 + A 
4. y = c1x + c2x ln(x) 
5. y = A sin(3x) + B cos(3x) 
6. y = c1e^(-3x) + c2e^(4x)

Learn how to find the differential equation by eliminating arbitrary constants from given functions. This example covers equations like y = Ax^2 + A and y = A sin(3x) + B cos(3x), including their classification by order, degree, and linearity.

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