Which one of the following equations is a homogeneous differential equation?

A differential equation of the form $M(x,y)dx + N(x,y)dy = 0$ is homogeneous if both $M(x,y)$ and $N(x,y)$ are homogeneous functions of the same degree.

A function is homogeneous of degree $n$ if replacing $x$ with $tx$ and $y$ with $ty$ allows factoring out $t^n$.

---

Option 1: $(4x + 5)dy + (3y - 4)dx = 0$

$M(x,y) = 3y - 4$

$N(x,y) = 4x + 5$

For $M(tx, ty) = 3(ty) - 4 = 3ty - 4$

The constant terms prevent factoring out any power of $t$.

Not homogeneous.

---

Option 2: $x^2y \, dx - (x^3 + y^3)dy = 0$

$M(x,y) = x^2y$

$N(x,y) = -(x^3 + y^3)$

For $M(tx, ty) = (tx)^2(ty)$

$= t^2x^2 \cdot ty$

$= t^3x^2y$

For $N(tx, ty) = -[(tx)^3 + (ty)^3]$

$= -(t^3x^3 + t^3y^3)$

$= t^3(-(x^3 + y^3))$

Both have degree 3. This is homogeneous.

---

Option 3: $(x^3 + 2y^2x)dy + 2xy \, dx = 0$

$M(x,y) = 2xy$

$N(x,y) = x^3 + 2y^2x$

For $M(tx, ty) = 2(tx)(ty)$

$= t^2(2xy)$

Degree 2.

For $N(tx, ty) = (tx)^3 + 2(ty)^2(tx)$

$= t^3x^3 + t^3(2y^2x)$

$= t^3(x^3 + 2y^2x)$

Degree 3.

Different degrees. Not homogeneous.

---

Option 4: $x^2 dy - y^2 dx = \sqrt{x^2 + y^2} \, dx$

Rearranging: $x^2 dy = (y^2 + \sqrt{x^2 + y^2})dx$

$M(x,y) = y^2 + \sqrt{x^2 + y^2}$

$N(x,y) = x^2$

For the term $\sqrt{x^2 + y^2}$:

$\sqrt{(tx)^2 + (ty)^2} = \sqrt{t^2x^2 + t^2y^2} = t\sqrt{x^2 + y^2}$

The term $y^2$ has degree 2 while $\sqrt{x^2 + y^2}$ has degree 1.

Different degree terms in the same function. Not homogeneous.

---

Option 2 is the homogeneous differential equation because both $M(x,y)$ and $N(x,y)$ are homogeneous functions of degree 3.