GATE 2026 CE (CE1) – Question 37

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

An ordinary differential equation is given below.

$$x^2\frac{d^2y}{dx^2} = 6y$$

Considering $a$ and $b$ as arbitrary constants, the general solution of the equation is

$y(x) = ax^3 + \dfrac{b}{x^2}$
$y(x) = ax^2 + \dfrac{b}{x^3}$
$y(x) = ax^2 + b\ln x$
$y(x) = ax^3 + b\ln x$

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Correct answer: (A) $y(x) = ax^3 + \dfrac{b}{x^2}$

Explanation

This is an Euler-Cauchy equation. Try $y = x^m$, so $m(m - 1) = 6$, which gives $m^2 - m - 6 = 0$ and $m = 3$ or $m = -2$. The general solution is $y = ax^3 + bx^{-2}$.