Differential Equations
Linear Differential Equation
Grade 12
Question:
<p>Given <br>\( x\dfrac{dy}{dx} + 2y = x^2 \)<br>If \( y(1) = 1 \), find \( y\!\left(\dfrac{1}{2}\right) \).</p>
<p>\(\dfrac{49}{16}\)</p>
<p>\(\dfrac{45}{16}\)</p>
<p>\(\dfrac{1}{4}\)</p>
<p>\(\dfrac{3}{4}\)</p>
Step-by-Step Solution
Key Concept: Recognize this as a linear first-order differential equation of the form dy/dx + P(x)y = Q(x). Divide by x to get standard form, then use integrating factor e^(∫P(x)dx) to solve.
<p><strong>Step 1:</strong> Rewrite in standard form by dividing by x:</p><p>dy/dx + (2/x)y = x</p><p><strong>Step 2:</strong> Find integrating factor: μ(x) = e^(∫2/x dx) = e^(2ln|x|) = x²</p><p><strong>Step 3:</strong> Multiply both sides by x²:</p><p>x²(dy/dx) + 2xy = x³</p><p>This gives: d/dx(x²y) = x³</p><p><strong>Step 4:</strong> Integrate both sides:</p><p>x²y = ∫x³ dx = x⁴/4 + C</p><p>Therefore: y = x²/4 + C/x²</p><p><strong>Step 5:</strong> Apply initial condition y(1) = 1:</p><p>1 = 1/4 + C ⟹ C = 3/4</p><p>So: y = x²/4 + 3/(4x²)</p><p><strong>Step 6:</strong> Find y(1/2):</p><p>y(1/2) = (1/4)·(1/4) + 3/4·(1/(1/4)) = 1/16 + 3 = 49/16</p><p>∴ Answer: A</p>
Correct Answer: A