Differential Equations
Complex ODE — substitution
Grade Class 12
Question:
<p>\\(y'-y\\tan x=2x\\sec x\\), \\(y(0)=0\\). Find \\(y(\\pi/4)\\).</p>
<span>\(\frac{\pi^2}{8\sqrt{2}}\)</span>
<span>\(\frac{\pi^2}{4\sqrt{2}}\)</span>
<span>\(\frac{\pi}{4\sqrt{2}}\)</span>
<span>\(\frac{\pi^2}{4}\)</span>
Step-by-Step Solution
Key Concept: IF = cos x. Solve to get y = x^2sec x.
<div class='solution'><p>IF $=e^{-\int\tan x\,dx}=\cos x$. $\frac{d}{dx}(y\cos x)=2x\cos x\cdot\sec x=2x$. $y\cos x=x^2+C$. $y(0)=0$: $C=0$. $y=x^2\sec x$. $y(\pi/4)=(\pi/4)^2\cdot\sec(\pi/4)=\dfrac{\pi^2}{16}\cdot\sqrt{2}=\dfrac{\pi^2\sqrt{2}}{16}=\dfrac{\pi^2}{8\sqrt{2}}$. <strong>Answer: (1)</strong> $\dfrac{\pi^2}{8\sqrt{2}}$.</p></div>
Correct Answer: 1