Question

If \(y = y(x)\) is the solution of the differential equation,
\(x\dfrac{dy}{dx} + 2y = x^2\), satisfying \(y(1) = 1\), then \(y\left(\frac12\right)\) is equal to :

• A. \(\frac{7}{64}\)
• B. \(\frac{49}{16}\)
• C. \(\frac14\)
• D. \(\frac{13}{16}\)

Hint:

Divide by \(x\) to get a linear equation with integrating factor \(x^2\); then use \(y(1) = 1\).

Answer 1

The Correct Option is B

\(\frac{dy}{dx} + \frac2x y = x\). Integrating factor \(= e^{\int 2/x\,dx} = x^2\).
\(\frac{d}{dx}(x^2y) = x^3 \Rightarrow x^2y = \frac{x^4}{4} + C\).
\(y(1) = 1 \Rightarrow 1 = \frac14 + C \Rightarrow C = \frac34\), so \(y = \frac{x^2}{4} + \frac{3}{4x^2}\).
\(y\left(\frac12\right) = \frac{1}{16} + \frac{3}{4 \cdot \frac14} = \frac1{16} + 3 = \frac{49}{16}\).
Answer: (2) \(\frac{49}{16}\)