https://file.aiquickdraw.com/mathbot/file/1724066636936upz78dy0.jpg

The image contains a mathematical expression, and it seems that the question is asking whether this expression is true or false. Let's break down the expression and analyze it step by step.

### Expression Analysis:

The equation appears to involve a differential equation along with some algebraic terms. Based on the image, the expression can be interpreted as follows:

\[
\text{The true solution of the equation} \quad \frac{d^2y}{dx^2} - \frac{dy}{dx} = 3y(x, y) = \frac{3y}{x^2} = 2 \quad \text{is} \quad y = \frac{1}{3} - \frac{1}{3}x^2
\]

To evaluate if the equation is true or false, we need to consider the following steps:

1. **Differential Equation Interpretation**:
   - The left-hand side of the equation involves the second derivative of \( y \) with respect to \( x \) minus the first derivative of \( y \) with respect to \( x \).
   - The expression on the right-hand side is \( 3y(x, y) \), and it is stated that \( 3y(x, y) = \frac{3y}{x^2} = 2 \).

2. **Given Solution**:
   - The proposed solution is \( y = \frac{1}{3} - \frac{1}{3}x^2 \).

### Verification:

Let's verify if the proposed solution satisfies the differential equation:

1. **First derivative**:
   \[
   \frac{dy}{dx} = -\frac{2}{3}x
   \]

2. **Second derivative**:
   \[
   \frac{d^2y}{dx^2} = -\frac{2}{3}
   \]

3. **Substitute into the Differential Equation**:
   \[
   \frac{d^2y}{dx^2} - \frac{dy}{dx} = -\frac{2}{3} - \left(-\frac{2}{3}x\right) = -\frac{2}{3} + \frac{2}{3}x
   \]
   Compare this with the right-hand side of the original equation \( 3y(x, y) = \frac{3y}{x^2} = 2 \).

Given this analysis, the original differential equation and the proposed solution don't seem to satisfy each other based on the general form of the solution.

### Conclusion:
The expression appears to be **false** based on the mathematical steps we applied.

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Would you like more details on this process or have any questions?

Here are some related questions you might find useful:

1. What are the general methods for solving second-order differential equations?
2. How do you check if a function is a solution to a differential equation?
3. What are some common mistakes when solving differential equations?
4. How do boundary conditions affect the solutions of differential equations?
5. What are the different types of differential equations and their applications?

**Tip:** Always verify each step when solving differential equations, especially when checking if a function is a solution to a given equation.

Explore the analysis of a differential equation solution to determine its correctness. This detailed breakdown covers differential equation solving methods and solution verification techniques. Suitable for advanced college-level students and mathematicians.

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