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Find the differential equation that is equal to 0 and after determine the order degree and linearity

From the image you've provided, you've calculated the first and second derivatives of a function \( y = Ax^2 + A \) and derived an equation for \( A \). Let's use these to form a differential equation that is equal to 0.

### Steps and Calculations
1. Given the function:
   \[
   y = Ax^2 + A
   \]
2. First derivative:
   \[
   y' = 2Ax
   \]
3. Second derivative:
   \[
   y'' = 2A
   \]

4. You've already solved for \( A \) from \( y'' \):
   \[
   A = \frac{y''}{2}
   \]

5. Substituting \( A \) back into the original equation:
   \[
   y = \left(\frac{y''}{2}\right)x^2 + \frac{y''}{2}
   \]

This can be simplified and rearranged to form a differential equation by setting it equal to 0:
   \[
   y - \frac{y'' x^2}{2} - \frac{y''}{2} = 0
   \]

### Differential Equation
The differential equation is:
   \[
   y - \frac{y'' x^2 + y''}{2} = 0 \quad \text{or} \quad 2y - y''x^2 - y'' = 0
   \]

### Order, Degree, and Linearity
- **Order**: The order of a differential equation is the order of the highest derivative it contains. Here, the highest derivative is \( y'' \), so the order is **2**.
- **Degree**: The degree of a differential equation is the highest power of the highest derivative. Since \( y'' \) appears linearly (not raised to any power greater than 1), the degree is **1**.
- **Linearity**: A differential equation is linear if it can be expressed as a linear combination of the derivatives of \( y \) (including \( y \) itself) equated to a function of \( x \). This equation, \( 2y - y''x^2 - y'' = 0 \), is linear as it does not contain any products or nonlinear functions of \( y \), \( y' \), or \( y'' \).

Would you like any further details or have any other questions about this solution?

### Further Discussion Points
1. How can the coefficients in a differential equation affect its general solution?
2. What methods are available to solve second-order linear differential equations?
3. How do initial conditions influence the solution to a differential equation?
4. Can you provide examples of physical phenomena modeled by second-order linear differential equations?
5. What are the implications of a non-constant coefficient (like \( x^2 \) in this case) in a differential equation?

### Tip
Remember, the characteristics of the differential equation (order, degree, linearity) help determine the methods suitable for finding its solutions and understanding its behavior.

Find the differential equation that is equal to 0 and after determine the order, degree, and linearity.

Solve for the differential equation starting from y = Ax^2 + A and determining the order, degree, and linearity of the result. The solution involves taking the first and second derivatives, substituting values, and analyzing the differential equation.

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