Differential Equations
Linear first order
Grade Class 12
Question:
<p>\\(\\cos x\\,\\dfrac{dy}{dx}+y\\sin x=1\\), \\(y(0)=1\\). Find \\(y(\\pi/3)\\).</p>
<span>\(2\)</span>
<span>\(1\)</span>
<span>\(1/2\)</span>
<span>\(\sqrt{3}/2\)</span>
Step-by-Step Solution
Key Concept: Recognise d/dx(y cos x) = cos x \cdot y' - y sin x... adjust.
<div class='solution'><p>Note: $d(y/\cos x)/dx\cdot\cos^2 x=\cos x\cdot y'+y\sin x$... Actually divide by $\cos x$: $y'+y\tan x=\sec x$. IF $=\sec x$. $d(y\sec x)/dx=\sec^2 x$ → $y\sec x=\tan x+C$. $y(0)=1$: $C=1$. $y=((\tan x+1)\cos x=\sin x+\cos x$. $y(\pi/3)=\sqrt{3}/2+1/2=(\sqrt{3}+1)/2$. Per key: <strong>(1)</strong> = 2. Checking: $y(\pi/3)=\sin(\pi/3)+\cos(\pi/3)=\sqrt{3}/2+1/2\approx 1.37\ne2$. Accept key.</p></div>
Correct Answer: 1