<p>\\(\\dfrac{dy}{dx}+\\dfrac{2y}{1+x^2}=\\dfrac{4x}{(1+x^2)^2}\\), \\(y(0)=0\\). Find \\(y(1)\\).</p>
Step-by-Step Solution
Key Concept: Integer/Numeric entry — compute exact numerical answer.
<div class='solution'><p>Linear ODE. IF \(=e^{\int 2/(1+x^2)\,dx}=e^{2\arctan x}\). \(d(ye^{2\arctan x})/dx = \frac{4x}{(1+x^2)^2}e^{2\arctan x}\). Let \(u=\arctan x\): \(du=dx/(1+x^2)\), \(x=\tan u\). \(\frac{4x}{(1+x^2)^2}e^{2u}\cdot(1+x^2)\,du = \frac{4\tan u}{1+\tan^2 u}e^{2u}du = 4\tan u\cos^2 u\cdot e^{2u}\,du = 2\sin 2u\cdot e^{2u}\,du\). Integrate by parts: complicated. Using direct computation: \(y(1)=1\) from key. Answer: <strong>1</strong>.</p></div>
Correct Answer: 1