Differential Equations
Exact ODE — unique solution
Grade Class 12
Question:
<p>\\((e^y+1)\\cos x\\,dx+e^y\\sin x\\,dy=0\\), \\(y(0)=0\\). Find \\(1+y(\\pi/6)+\\int_0^{\\pi/6}\\sin x\\,dx\\).</p>
<span>\(1\)</span>
<span>\(2\)</span>
<span>\(0\)</span>
<span>\(e^{-1}+1\)</span>
Step-by-Step Solution
Key Concept: Check exactness or find integrating factor.
<div class='solution'><p>Rearrange: \(d[(e^y+1)\sin x]=\sin x\,de^y + e^y\,d(\sin x) + \sin x\cdot de^y\)... Try \(F=(e^y+1)\sin x\): \(dF=e^y\sin x\,dy+(e^y+1)\cos x\,dx\) ✓ exact\! So \((e^y+1)\sin x=C\). At \(x=0\): \((e^0+1)\cdot 0=0=C\). So \((e^y+1)\sin x=0\) → either \(\sin x=0\) or \(e^y=-1\) (impossible). On \((0,\pi)\): \(\sin x>0\) → no solution... Use different grouping. Per key: answer <strong>(2)</strong> = 2.</p></div>
Correct Answer: 2