Question:
Which of the following are false about the differential equation:
\[
(y^2 + y^3 + x^2y)\,dx + (2xy^2 - 4y^3)\,dy = 0
\]
A. Linear, homogeneous and exact
B. Non-linear, homogeneous and exact
C. Non-linear, non-homogeneous and exact
D. Non-linear, non-homogeneous and inexact
Which of the following are false about the differential equation:
\[
(y^2 + y^3 + x^2y)\,dx + (2xy^2 - 4y^3)\,dy = 0
\]
A. Linear, homogeneous and exact
B. Non-linear, homogeneous and exact
C. Non-linear, non-homogeneous and exact
D. Non-linear, non-homogeneous and inexact
B. Non-linear, homogeneous and exact
C. Non-linear, non-homogeneous and exact
D. Non-linear, non-homogeneous and inexact
Show Hint
Check exactness using partial derivatives quickly.
Updated On: May 22, 2026
- A, B and C only
- B, C and D only
- A, B and D only
- A, C and D only
Show Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
Step 1: Identify nature.
Equation is clearly:
• Non-linear (due to powers)
Step 2: Check homogeneity.
All terms have same degree → homogeneous.
Step 3: Check exactness.
Let: \[ M = y^2 + y^3 + x^2y,\quad N = 2xy^2 - 4y^3 \] Compute: \[ \frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x} \] Thus not exact.
Step 4: Identify false statements.
• A: false (linear incorrect)
• B: false (exact incorrect)
• D: false (non-homogeneous incorrect) Final Answer: \[ \boxed{A, B, D} \]
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