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.