12. What is the meaning of the differential equation

dy

dx

=

x

y

, and how do you interpret it?

The differential equation \(\frac{dy}{dx} = \frac{x}{y}\) describes the relationship between the independent variable \(x\) and the dependent variable \(y\). Here's what it means and how you can interpret it:

### 1. **Meaning:**
   - The equation \(\frac{dy}{dx} = \frac{x}{y}\) expresses that the rate of change of \(y\) with respect to \(x\) is proportional to the ratio of \(x\) to \(y\).

### 2. **Interpretation:**
   - **Slope Field Interpretation:** For any point \((x, y)\) on the plane, the value \(\frac{x}{y}\) gives the slope of the tangent to the curve \(y(x)\) at that point. This means that the direction of the curve at any given point is determined by the ratio of the \(x\)-coordinate to the \(y\)-coordinate at that point.
   - **Homogeneous Equation:** This differential equation is homogeneous because it can be written in the form \(\frac{dy}{dx} = f\left(\frac{x}{y}\right)\), which implies that the equation has certain symmetry properties. In this case, it depends only on the ratio \(\frac{x}{y}\).
   - **Separable Equation:** The equation can be rewritten as \(y \, dy = x \, dx\), making it separable. This means you can solve it by integrating both sides separately.

### 3. **Qualitative Analysis:**
   - **Behavior of Solutions:** The solutions of this differential equation typically describe curves where the slope at any point is directly related to how \(x\) compares to \(y\). When \(x > y\), the slope is positive, and when \(x < y\), the slope is negative. When \(x = y\), the slope is 1.
   - **Symmetry:** Due to the homogeneous nature, if \(y(x)\) is a solution, then \(c \cdot y(x)\) is also a solution for any constant \(c\). This indicates scaling properties of the solution curves.

### 4. **Solutions:**
   - **General Solution:** By separating variables and integrating, the general solution to the equation is \(y^2 = x^2 + C\), where \(C\) is an arbitrary constant. This represents a family of hyperbolas.
   - **Special Case:** If \(C = 0\), the solution is \(y = \pm x\), which are straight lines through the origin with slopes of \(\pm 1\).

### 5. **Geometric Interpretation:**
   - The equation describes curves where the rate of change of \(y\) with respect to \(x\) is governed by how \(x\) and \(y\) relate to each other through the simple ratio \(\frac{x}{y}\). The general solution \(y^2 = x^2 + C\) describes hyperbolas centered at the origin.

### Summary:
The differential equation \(\frac{dy}{dx} = \frac{x}{y}\) is a first-order, homogeneous, and separable ordinary differential equation. It describes how \(y\) changes with \(x\) based on their ratio, leading to a family of hyperbolic solutions or straight lines when \(C = 0\).

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Would you like further details or have any specific questions?

### 8 Related Questions:
1. How do you solve a homogeneous differential equation like \(\frac{dy}{dx} = \frac{x}{y}\)?
2. What does the solution \(y^2 = x^2 + C\) represent geometrically?
3. Can you show how to solve \(\frac{dy}{dx} = \frac{x}{y}\) step by step?
4. What are the characteristics of the solution curves for \(\frac{dy}{dx} = \frac{x}{y}\)?
5. How does the symmetry of the differential equation \(\frac{dy}{dx} = \frac{x}{y}\) influence its solutions?
6. What are some real-world scenarios where the differential equation \(\frac{dy}{dx} = \frac{x}{y}\) might arise?
7. How does the behavior of solutions change with different initial conditions in \(\frac{dy}{dx} = \frac{x}{y}\)?
8. What other methods, besides separation of variables, could be used to solve this equation?

### Tip:
When solving homogeneous differential equations, look for opportunities to simplify the equation by using substitution or separation of variables. Recognizing the symmetry and scaling properties can also guide you to the correct solution.

Explore the meaning and interpretation of the differential equation \(\frac{dy}{dx} = \frac{x}{y}\). Learn how to interpret slope field, homogeneous nature, separable form, qualitative analysis, and geometric interpretation of solutions. Suitable for advanced high school students and those interested in differential equations.

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