Which of the following is a homogenous differential equation ?

Options

A. `(4x+6y+5)dy-(3y+2x+4)dx=0`
B. `(xy)dx-(x^(3)+y^(3))dy=0`
C. `(x^(3)+2y^(2))dx+2xydy=0`
D. `y^(2)dx+(x^(2)-xy-y^(2))dy=0`

Explanation

To determine which of the given options is a homogeneous differential equation, we will analyze each option based on the definition of a homogeneous differential equation. A differential equation is considered homogeneous if it satisfies the condition:

\[ f(zx, zy) = z^n f(x, y) \]

for some integer \( n \).

Let's analyze each option step by step.

### Step 1: Analyze the first option

Given:
\[ 4x + 6y + 5 \, dy - (3y + 2x + 4) \, dx = 0 \]

We can rearrange it to find \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{3y + 2x + 4}{4x + 6y + 5} \]

Now, we will check if it satisfies the homogeneous condition:
- Substitute \( zx \) for \( x \) and \( zy \) for \( y \):
\[ f(zx, zy) = \frac{3(zy) + 2(zx) + 4}{4(zx) + 6(zy) + 5} = \frac{3zy + 2zx + 4}{4zx + 6zy + 5} \]

This does not simplify to \( z^n f(x, y) \) for any integer \( n \) because the terms do not match in order.

**Conclusion**: This is **not a homogeneous equation**.

### Step 2: Analyze the second option

Given:
\[ xy \, dx - (x^3 + y^3) \, dy = 0 \]

Rearranging gives:
\[ \frac{dy}{dx} = \frac{xy}{x^3 + y^3} \]

Now, check the homogeneous condition:
- Substitute \( zx \) for \( x \) and \( zy \) for \( y \):
\[ f(zx, zy) = \frac{(zx)(zy)}{(zx)^3 + (zy)^3} = \frac{z^2xy}{z^3(x^3 + y^3)} \]

This does not simplify to \( z^n f(x, y) \) because the degrees of the numerator and denominator do not match.

**Conclusion**: This is **not a homogeneous equation**.

### Step 3: Analyze the third option

Given:
\[ (x^3 + 2y^2) \, dx + (2xy) \, dy = 0 \]

Rearranging gives:
\[ \frac{dy}{dx} = -\frac{x^3 + 2y^2}{2xy} \]

Now, check the homogeneous condition:
- Substitute \( zx \) for \( x \) and \( zy \) for \( y \):
\[ f(zx, zy) = -\frac{(zx)^3 + 2(zy)^2}{2(zx)(zy)} = -\frac{z^3x^3 + 2z^2y^2}{2z^2xy} \]

This does not simplify to \( z^n f(x, y) \) for any integer \( n \).

**Conclusion**: This is **not a homogeneous equation**.

### Step 4: Analyze the fourth option

Given:
\[ y^2 \, dx + (x^2 - xy - y^2) \, dy = 0 \]

Rearranging gives:
\[ \frac{dy}{dx} = -\frac{y^2}{x^2 - xy - y^2} \]

Now, check the homogeneous condition:
- Substitute \( zx \) for \( x \) and \( zy \) for \( y \):
\[ f(zx, zy) = -\frac{(zy)^2}{(zx)^2 - (zx)(zy) - (zy)^2} = -\frac{z^2y^2}{z^2x^2 - z^2xy - z^2y^2} \]

This simplifies to:
\[ -\frac{y^2}{x^2 - xy - y^2} \]

This satisfies the condition \( f(zx, zy) = z^0 f(x, y) \).

**Conclusion**: This is a **homogeneous equation**.

### Final Answer
The correct option is **Option 4**, which is a homogeneous differential equation.

---