GATE 2023 EE – Question 34

Engineering Mathematics · Differential Equations: First order equations, Higher order linear differential equations, Variation of parameters, Cauchy and Euler equations, Partial differential equations, Separation of variables · 1 mark · Numerical answer

In the following differential equation, the numerically obtained value of $y(t)$, at $t=1$, is _______________ (Round off to 2 decimal places).

$$\frac{dy}{dt}=\frac{e^{-\alpha t}}{2+\alpha t},\qquad\alpha=0.01\ \text{and}\ y(0)=0$$

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Correct answer: 0.49 to 0.51

Explanation

$y(1)=\int_0^1\dfrac{e^{-0.01t}}{2+0.01t}dt$. The integrand is 0.5 at $t=0$, 0.4963 at $t=0.5$ and 0.4926 at $t=1$. Simpson's rule gives $\dfrac16(0.5+4(0.4963)+0.4926)=0.4963\approx0.50$.