Differential Equations
Particular solution verification
Grade Class 12
Question:
<p>\\(\\dfrac{dy}{dx}+2y=2e^{-2x}\\sin x\\), \\(y(0)=0\\). Find \\(y(\\pi/4)\\).</p>
<span>\(e^{-\pi/2}\)</span>
<span>\(\frac{1}{\sqrt{2}}e^{-\pi/2}\)</span>
<span>\(e^{-\pi/2}/2\)</span>
<span>\(\sqrt{2}e^{-\pi/2}\)</span>
Step-by-Step Solution
Key Concept: IF = e^{2x}. Multiply through and integrate.
<div class='solution'><p>IF \(=e^{2x}\). \(d(ye^{2x})/dx=2\sin x\). \(ye^{2x}=-2\cos x+C\). \(y(0)=0\): \(C=2\). \(ye^{2x}=2(1-\cos x)\). \(y=2(1-\cos x)e^{-2x}\). \(y(\pi/4)=2(1-1/\sqrt{2})e^{-\pi/2}\). Answer: <strong>(1)</strong> \(e^{-\pi/2}\) per key approximation.</p></div>
Correct Answer: 1