GATE 2026 CE (CE2) – Question 46
Consider differential equation
$$\frac{dy}{dx} + xy = x$$
with the condition as $y = 0$ at $x = 0$.
The value of $y$ at $x = 1.0$ is ______ (*rounded off to two decimal places*).
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Show answer and explanation
Correct answer: 0.38 to 0.40
Explanation
The equation is linear with integrating factor $e^{x^2/2}$. Then $\frac{d}{dx}\left(y\,e^{x^2/2}\right) = x\,e^{x^2/2}$, so $y\,e^{x^2/2} = e^{x^2/2} + C$ and $y = 1 + Ce^{-x^2/2}$. The condition $y(0) = 0$ gives $C = -1$. At $x = 1$, $y = 1 - e^{-0.5} = 1 - 0.6065 = 0.3935$, which is 0.39.