Differential Equations
Linear differential equations
Grade None
Question:
<p>The solution of the differential equation \(x\dfrac{dy}{dx} + 2y = x^2\;(x \neq 0)\) with \(y(1) = 1\), is:</p>
<p>\(y = \dfrac{4}{5}x^3 + \dfrac{1}{5x^2}\)</p>
<p>\(y = \dfrac{x^3}{5} + \dfrac{1}{5x^2}\)</p>
<p>\(y = \dfrac{x^2}{4} + \dfrac{3}{4x^2}\)</p>
<p>\(y = \dfrac{3}{4}x^2 + \dfrac{1}{4x^2}\)</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). Use integrating factor μ(x) = e^(∫P(x)dx) to convert it into an exact differential that can be integrated directly.
<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 is d/dx(x²y) = x³</p><p><strong>Step 4:</strong> Integrate both sides:</p><p>x²y = ∫x³ dx = x⁴/4 + C</p><p><strong>Step 5:</strong> Apply initial condition y(1) = 1:</p><p>(1)²(1) = (1)⁴/4 + C</p><p>1 = 1/4 + C</p><p>C = 3/4</p><p><strong>Step 6:</strong> General solution:</p><p>x²y = x⁴/4 + 3/4</p><p>y = x²/4 + 3/(4x²)</p><p>∴ Answer: C</p>
Correct Answer: C