GATE 2023 CE (CE1) – Question 30
In the differential equation $\frac{dy}{dx} + \alpha xy = 0$, $\alpha$ is a positive constant. If $y = 1.0$ at $x = 0.0$, and $y = 0.8$ at $x = 1.0$, the value of $\alpha$ is _______________ (rounded off to three decimal places).
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Correct answer: 0.436 to 0.456
Explanation
Separate the variables: $\frac{dy}{y} = -\alpha x\,dx$, so $y = y_0e^{-\alpha x^2/2}$. With $y_0 = 1$ and $y(1) = 0.8$: $0.8 = e^{-\alpha/2}$, so $\alpha = -2\ln 0.8 = 0.446$.