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.