GATE 2026 AE – Question 49

Engineering Mathematics · Differential Equations: First order linear ODEs and higher order linear ODEs with constant coefficients · 2 marks · Numerical answer

Consider the differential equation with the initial conditions given below. If $y(x)$ is the solution of the equation, the value of the slope, $\frac{dy}{dx}$, at $x = \ln(2)$ is ________ (rounded off to three decimal places).

$\frac{d^2y}{dx^2} + 2\frac{dy}{dx} + y = 0$ with $y|_{x=0} = 0$ and $\frac{dy}{dx}\Big|_{x=0} = 1$

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Correct answer: 0.143 to 0.163

Explanation

The characteristic equation $(m + 1)^2 = 0$ has a repeated root, so $y = (C_1 + C_2 x)e^{-x}$. From $y(0) = 0$, $C_1 = 0$, and from $y'(0) = C_2 = 1$ we get $y = xe^{-x}$ and $y' = (1 - x)e^{-x}$. At $x = \ln 2$: $y' = (1 - 0.6931) \times 0.5 = 0.153$.