<p>If \(\frac{dy}{dx} + y\tan x = \sin 2x\) and \(y(0) = 1\), then \(y(\pi)\) is equal to</p>
Step-by-Step Solution
Key Concept: This is a first-order linear differential equation of the form dy/dx + P(x)y = Q(x). Identify the integrating factor e^∫P(x)dx = e^∫tan x dx = e^ln|sec x| = sec x, then multiply and integrate to find y(x).
<p><strong>Step 1:</strong> Identify the standard form. We have dy/dx + y tan x = sin 2x, so P(x) = tan x and Q(x) = sin 2x.</p><p><strong>Step 2:</strong> Find the integrating factor: I.F. = e^∫tan x dx = e^ln|sec x| = sec x</p><p><strong>Step 3:</strong> Multiply both sides by sec x:</p><p>sec x · dy/dx + sec x · y tan x = sec x · sin 2x</p><p>d/dx(y sec x) = sec x · 2sin x cos x = 2sin x · sec x · cos x = 2sin x</p><p><strong>Step 4:</strong> Integrate both sides:</p><p>y sec x = ∫2sin x dx = -2cos x + C</p><p><strong>Step 5:</strong> Apply initial condition y(0) = 1:</p><p>1 · sec 0 = -2cos 0 + C</p><p>1 = -2 + C ⟹ C = 3</p><p><strong>Step 6:</strong> Find y at x = π:</p><p>y sec π = -2cos π + 3</p><p>y · (-1) = -2(-1) + 3 = 2 + 3 = 5</p><p>y(π) = -5</p><p>∴ Answer: B (y(π) = -5)</p>
Correct Answer: B