GATE 2026 CH – Question 62

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

Consider a differential equation:

$x^2\frac{d^2y}{dx^2} + 3x\frac{dy}{dx} - 3y = 0$

$y = 3$, $\frac{dy}{dx} = -5$ at $x = 1$

The value of $y$ at $x = 2$ is ______ (rounded off to two decimal places).

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Correct answer: 2.24 to 2.26

Explanation

This is a Cauchy-Euler equation. Put $y = x^m$: $m(m - 1) + 3m - 3 = 0$, that is $m^2 + 2m - 3 = 0$, so $m = 1$ or $m = -3$. Then $y = ax + bx^{-3}$. At $x = 1$: $a + b = 3$ and $y' = a - 3b = -5$, so $4b = 8$, $b = 2$ and $a = 1$. At $x = 2$: $y = 2 + \frac{2}{8} = 2.25$.