how can you identify if the equation is homogeneous and exact DE

To determine whether a first-order differential equation (DE) is **homogeneous** or **exact**, we follow specific criteria for each.

### 1. **Homogeneous Differential Equation**
A differential equation of the form:
\[
M(x, y) \, dx + N(x, y) \, dy = 0
\]
is **homogeneous** if both functions \( M(x, y) \) and \( N(x, y) \) are homogeneous of the same degree. That is, when each term in both \( M(x, y) \) and \( N(x, y) \) can be rewritten as a function of \( \frac{y}{x} \).

#### Steps to Identify Homogeneity:
- Replace \( x \) by \( \lambda x \) and \( y \) by \( \lambda y \) in \( M(x, y) \) and \( N(x, y) \).
- If both \( M(\lambda x, \lambda y) \) and \( N(\lambda x, \lambda y) \) can be factored as \( \lambda^k M(x, y) \) and \( \lambda^k N(x, y) \), where \( k \) is the same for both, then the DE is homogeneous of degree \( k \).

#### Example:
\[
(3x^2 + 2xy) \, dx + (2x^2 + y^2) \, dy = 0
\]
- Replacing \( x \) by \( \lambda x \) and \( y \) by \( \lambda y \):
\[
M(\lambda x, \lambda y) = 3(\lambda x)^2 + 2(\lambda x)(\lambda y) = \lambda^2 (3x^2 + 2xy)
\]
\[
N(\lambda x, \lambda y) = 2(\lambda x)^2 + (\lambda y)^2 = \lambda^2 (2x^2 + y^2)
\]
Both are homogeneous of degree 2, so the equation is homogeneous.

---

### 2. **Exact Differential Equation**
A differential equation of the form:
\[
M(x, y) \, dx + N(x, y) \, dy = 0
\]
is **exact** if there exists a function \( \psi(x, y) \) such that:
\[
\frac{\partial \psi}{\partial x} = M(x, y) \quad \text{and} \quad \frac{\partial \psi}{\partial y} = N(x, y)
\]

#### Condition for Exactness:
To check if the DE is exact, verify that the following condition holds:
\[
\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}
\]
If this equality is true, then the DE is exact, and there exists a potential function \( \psi(x, y) \) whose total differential leads to the given equation.

#### Example:
Consider the equation:
\[
(2xy + y^2) \, dx + (x^2 + 2xy) \, dy = 0
\]
- Compute \( \frac{\partial M}{\partial y} \) and \( \frac{\partial N}{\partial x} \):
\[
\frac{\partial M}{\partial y} = 2x + 2y, \quad \frac{\partial N}{\partial x} = 2x + 2y
\]
Since \( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \), the equation is exact.

---

### Summary of Steps:

1. **Homogeneous DE**:
   - Check if \( M(x, y) \) and \( N(x, y) \) are homogeneous of the same degree.

2. **Exact DE**:
   - Verify that \( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \).

---

Would you like more details on a specific example or method? Here are 5 related questions:

1. How do you solve a homogeneous differential equation once identified?
2. What is the integrating factor if a DE is not exact?
3. How do you find the potential function for an exact DE?
4. Can a DE be both homogeneous and exact? How would you approach such a case?
5. What are some common methods to test for non-exact differential equations?

**Tip:** If a differential equation is not exact, you can often multiply by an integrating factor to make it exact.

Learn how to determine whether a differential equation is homogeneous or exact. This guide covers key concepts, including homogeneous and exact differential equations, relevant formulas, and step-by-step examples. Suitable for first-year calculus students.

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