Differential Equations
Exact ODE
Grade Class 12
Question:
<p>\\((3x^2y+y^3)\\,dx+(x^3+3xy^2)\\,dy=0\\). General solution:</p>
<span>\(x^3y + xy^3 = C\)</span>
<span>\(x^2y^2 + y^4 = C\)</span>
<span>\(x^3 + y^3 = C\)</span>
<span>\(x^4 + y^4 = C\)</span>
Step-by-Step Solution
Key Concept: Check exactness: \partialM/\partialy = \partialN/\partialx.
<div class='solution'><p>$\partial M/\partial y=3x^2+3y^2$, $\partial N/\partial x=3x^2+3y^2$ ✓ exact. $F_x=3x^2y+y^3$ → $F=x^3y+xy^3+g(y)$. $F_y=x^3+3xy^2+g'(y)=x^3+3xy^2$ → $g'=0$, $g=C$. Solution: $x^3y+xy^3=C$. <strong>Answer: (1)</strong>.</p></div>
Correct Answer: 1