GATE 2024 ME – Question 43

Engineering Mathematics · Differential Equations: First order and higher order linear equations, Euler-Cauchy equation · 2 marks · Numerical answer

If $x(t)$ satisfies the differential equation

$$t\frac{dx}{dt} + (t - x) = 0$$

subject to the condition $x(1) = 0$, then the value of $x(2)$ is ________ (*rounded off to 2 decimal places*).

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Correct answer: -1.40 to -1.38

Explanation

Divide by $t$: $\frac{dx}{dt} - \frac{x}{t} = -1$. The integrating factor is $\frac{1}{t}$, so $\frac{d}{dt}\left(\frac{x}{t}\right) = -\frac{1}{t}$ and $\frac{x}{t} = -\ln t + C$. The condition $x(1) = 0$ gives $C = 0$, so $x = -t\ln t$. At $t = 2$, $x = -2\ln 2 = -1.39$.