GATE 2025 CE (CE1) – Question 47

Engineering Mathematics · ODE: First order and higher order linear equations, Euler-Cauchy equations, initial and boundary value problems · 2 marks · Numerical answer

Let $y$ be the solution of the initial value problem $y'' + 0.8y' + 0.16y = 0$, where $y(0) = 3$ and $y'(0) = 4.5$. Then, $y(1)$ is equal to ______________________ (*rounded off to 1 decimal place*).

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Correct answer: 5.77 to 5.83

Explanation

The characteristic equation $r^2 + 0.8r + 0.16 = (r + 0.4)^2 = 0$ has a double root $r = -0.4$, so $y = (C_1 + C_2x)e^{-0.4x}$. From $y(0) = 3$, $C_1 = 3$. Then $y' = \left[C_2 - 0.4(C_1 + C_2x)\right]e^{-0.4x}$ and $y'(0) = C_2 - 1.2 = 4.5$, so $C_2 = 5.7$. At $x = 1$, $y = (3 + 5.7)e^{-0.4} = 8.7 \times 0.6703 = 5.83$, which is 5.8.