GATE 2025 AE – Question 43
Consider the ordinary differential equation: $\frac{1}{2}\frac{dy}{dx} + \frac{y}{x} = 1$. If $y = 2/3$ at $x = 1$, then the value of $y$ at $x = 3$ is ______ (rounded off to the nearest integer).
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Correct answer: 2
Explanation
Rewrite as $\frac{dy}{dx} + \frac{2}{x}y = 2$. The integrating factor is $x^2$, so $\frac{d}{dx}(x^2y) = 2x^2$ and $x^2y = \frac{2x^3}{3} + C$. At $x = 1$, $y = \frac{2}{3}$ gives $C = 0$. So $y = \frac{2x}{3}$ and $y(3) = 2$.