Differential Equations
Variable separable / linear ODE
Grade 12
Question:
<p>If \((2 + \sin x)\dfrac{dy}{dx} + (y+1)\cos x = 0\) and \(y(0) = 1\), then \(y\!\left(\dfrac{\pi}{2}\right)\) is equal to</p>
<p>\(-\dfrac{2}{3}\)</p>
<p>\(-\dfrac{1}{3}\)</p>
<p>\(\dfrac{4}{3}\)</p>
<p>\(\dfrac{1}{3}\)</p>
Step-by-Step Solution
Key Concept: Recognize this as a separable differential equation. Rearrange to isolate variables on opposite sides, then integrate both sides using substitution if needed.
<p><strong>Step 1:</strong> Rearrange the given equation as a separable form:</p><p>(2 + sin x)dy/dx = -(y+1)cos x</p><p>dy/(y+1) = -cos x/(2 + sin x) dx</p><p><strong>Step 2:</strong> Integrate both sides:</p><p>∫ dy/(y+1) = -∫ cos x/(2 + sin x) dx</p><p>ln|y+1| = -ln|2 + sin x| + C</p><p><strong>Step 3:</strong> Apply initial condition y(0) = 1:</p><p>ln|1+1| = -ln|2 + sin 0| + C</p><p>ln 2 = -ln 2 + C</p><p>C = 2ln 2</p><p><strong>Step 4:</strong> The solution is:</p><p>ln|y+1| = -ln|2 + sin x| + 2ln 2</p><p>ln|y+1| + ln|2 + sin x| = ln 4</p><p>(y+1)(2 + sin x) = 4</p><p><strong>Step 5:</strong> Evaluate at x = π/2:</p><p>(y + 1)(2 + sin π/2) = 4</p><p>(y + 1)(2 + 1) = 4</p><p>3(y + 1) = 4</p><p>y + 1 = 4/3</p><p>∴ y(π/2) = <strong>1/3</strong></p>
Correct Answer: D