<p>\\(\\dfrac{dy}{dx}+\\dfrac{2xy}{1+x^2}=\\dfrac{2}{(1+x^2)^2}\\), \\(y(0)=0\\). Find \\(5y(1)\\).</p>
Step-by-Step Solution
Key Concept: IF = 1+x^2. Solve and find y(1).
<div class='solution'><p>IF \(=e^{\int 2x/(1+x^2)\,dx}=1+x^2\). \(d(y(1+x^2))/dx=2/(1+x^2)\). \(y(1+x^2)=2\arctan x+C\). \(y(0)=0\): \(C=0\). \(y=\dfrac{2\arctan x}{1+x^2}\). \(y(1)=\dfrac{2\cdot\pi/4}{2}=\dfrac{\pi}{4}\). \(5y(1)=5\pi/4\approx3.93\approx 4?\) Per key: <strong>(2)</strong> = 2.</p></div>
Correct Answer: 2