<p>If \(\frac{dy}{dx} = y + 3\) and \(y(0) = 2\), then \(y(\ln 2)\) is equal to:</p>
Step-by-Step Solution
Key Concept: Recognize this as a first-order linear ODE of the form dy/dx - y = 3. Use separation of variables or integrating factor method to solve, then apply the initial condition y(0) = 2 to find the particular solution.
<p><strong>Step 1:</strong> Rewrite the ODE in standard form: dy/dx - y = 3</p><p><strong>Step 2:</strong> Use separation of variables. Rearrange: dy/(y + 3) = dx</p><p><strong>Step 3:</strong> Integrate both sides: ∫dy/(y + 3) = ∫dx → ln|y + 3| = x + C</p><p><strong>Step 4:</strong> Exponentiate: y + 3 = Ae^x, so y(x) = Ae^x - 3</p><p><strong>Step 5:</strong> Apply initial condition y(0) = 2: 2 = Ae^0 - 3 → 2 = A - 3 → A = 5</p><p><strong>Step 6:</strong> Particular solution: y(x) = 5e^x - 3</p><p><strong>Step 7:</strong> Evaluate at x = ln 2: y(ln 2) = 5e^(ln 2) - 3 = 5(2) - 3 = 10 - 3 = 7</p><p>∴ <strong>Answer: y(ln 2) = 7</strong></p>
Correct Answer: D