GATE 2023 ME – Question 49

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

Consider the second-order linear ordinary differential equation $x^2\frac{d^2y}{dx^2} + x\frac{dy}{dx} - y = 0$, $x \ge 1$ with the initial conditions $y(x = 1) = 6$, $\frac{dy}{dx}\Big|_{x=1} = 2$.
The value of $y$ at $x = 2$ equals _________.
(Answer in integer)

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Correct answer: 9

Explanation

This is a Cauchy-Euler equation. Putting $y = x^m$ gives $m(m - 1) + m - 1 = 0$, that is $m^2 = 1$, so $m = \pm 1$ and $y = ax + \frac{b}{x}$. The conditions are $a + b = 6$ and $y'(1) = a - b = 2$, so $a = 4$ and $b = 2$. At $x = 2$: $y = 4 \times 2 + \frac{2}{2} = 9$.