Multiple choice

The solution of the differential equation $\frac{dy}{dx} + \frac{y}{x} = x$, with the condition that y = 1 at x = 1, is

  1. $y = \frac{2}{3x^2} + \frac{x}{3}$
  2. $y = \frac{x}{2} + \frac{1}{2x}$
  3. $y = \frac{2}{3} + \frac{x}{3}$
  4. $y = \frac{2}{3x} + \frac{x^2}{3}$
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D Correct answer
Explanation

This is a linear differential equation dy/dx + (1/x)y = x. The integrating factor is e^(integral 1/x dx) = x. Multiplying by x gives d/dx(xy) = x^2. Integrating both sides gives xy = x^3/3 + C. Using y(1) = 1, we get 1 = 1/3 + C, so C = 2/3. Thus, xy = x^3/3 + 2/3, or y = x^2/3 + 2/(3x).

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