Question

The differential equation \[ (3x^2y + y^3)\,dx + (x^3 + 3xy^2)\,dy = 0 \] is

• A. homogeneous and exact
• B. neither separable nor exact
• C. exact and not homogeneous
• D. homogeneous and not exact

Hint:

For differential equations, always check homogeneity first by verifying the degree of terms, and then test for exactness using partial derivatives.

Answer 1

The Correct Option is A

Step 1: Identify functions.
Let $M = 3x^2y + y^3$ and $N = x^3 + 3xy^2$.
Step 2: Check for homogeneity.
Each term in $M$ and $N$ is of degree 3. Hence, both are homogeneous functions of degree 3.
Step 3: Check for exactness.
Compute partial derivatives: \[ \frac{\partial M}{\partial y} = 3x^2 + 3y^2, \quad \frac{\partial N}{\partial x} = 3x^2 + 3y^2 \] Since $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$, the equation is exact.
Step 4: Conclusion.
Therefore, the given differential equation is both homogeneous and exact.