Question:
If \( x^2 + xy^2 = c \), where \( c \in \mathbb{R} \), is the general solution of the exact differential equation
\[
M(x, y)\, dx + 2xy\, dy = 0,
\]
then \( M(1,1) \) is ............
If \( x^2 + xy^2 = c \), where \( c \in \mathbb{R} \), is the general solution of the exact differential equation
\[
M(x, y)\, dx + 2xy\, dy = 0,
\]
then \( M(1,1) \) is ............
Show Hint
To find \( M(x, y) \) in an exact differential equation, differentiate the given potential function and match coefficients with the differential form.
Updated On: Dec 6, 2025
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Correct Answer: 3
Solution and Explanation
Step 1: Differentiate the given equation.
Given \( x^2 + xy^2 = c \), differentiating both sides: \[ 2x\,dx + (y^2 + 2xy\,dy) = 0. \] So, \[ (2x + y^2)\,dx + 2xy\,dy = 0. \]
Step 2: Compare with given form.
Given equation: \( M(x, y)\,dx + 2xy\,dy = 0. \) Thus, \( M(x, y) = 2x + y^2. \)
Step 3: Evaluate at (1,1).
\[ M(1,1) = 2(1) + (1)^2 = 3. \]
Final Answer: \[ \boxed{3} \]
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