Differential Equations
Exact ODE — recognition and solution
Grade Class 12
Question:
<p>\\((y\\cos x+2xe^y)\\,dx+(\\sin x+x^2e^y-1)\\,dy=0\\). Select true:</p>
<span>\(\text{(A) Exact}\)</span>
<span>\(\text{(B) }\partial M/\partial y=\partial N/\partial x\)</span>
<span>\(\text{(C) }y\sin x+x^2e^y-y=C\)</span>
<span>\(\text{(D) Needs IF}\)</span>
Step-by-Step Solution
Key Concept: Check \partialM/\partialy and \partialN/\partialx.
<div class='solution'><p>M=$y\cos x+2xe^y$, N=$\sin x+x^2e^y-1$. $\partial M/\partial y = \cos x+2xe^y$. $\partial N/\partial x = \cos x+2xe^y$. Equal → exact ✓ (A),(B). $F_x = M$: $F = y\sin x + x^2e^y + g(y)$. $F_y = \sin x + x^2e^y + g'(y) = N = \sin x+x^2e^y-1$ → $g'(y)=-1$ → $g=-y$. Solution: $y\sin x+x^2e^y-y=C$ ✓ (C). (D) false — no IF needed. Per key: A,B.</p></div>
Correct Answer: A,B