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.

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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.

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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 $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$.

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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.