Solve the first-order differential equation dy/dx + (2/x) y = 4x with initial condition y(1) = 2.
Answer and explanation
Correct answer: C. y = x^2 + 1 / x^2
- Step 1: Integrating factor: I(x) = exp(integral 2/x dx) = exp(2 ln x) = x^2. Step 2. Multiply through: d/dx [y x^2] = 4x x^2 = 4x^3. Step 3. Integrate: y x^2 = x^4 + C => y = x^2 + C / x^2. Step 4. Apply y(1) = 2: 2 = 1 + C => C = 1. Therefore, y = x^2 + 1 / x^2. Answer: y = x^2 + 1 / x^2. Why not the others:* y = 2x^2 fails the differential equation check.
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