GATE 2024 CH – Question 52
Consider the ordinary differential equation $x^2\frac{d^2y}{dx^2} - x\frac{dy}{dx} - 3y = 0$, with the boundary conditions $y(x = 1) = 2$ and $y(x = 2) = 17/2$. The solution $y(x)$ at $x = 3/2$, rounded off to 2 decimal places, is ___________
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Correct answer: 4.02 to 4.06
Explanation
This is a Cauchy-Euler equation. Putting $y = x^m$ gives $m(m - 1) - m - 3 = 0$, that is $m^2 - 2m - 3 = 0$, so $m = 3$ or $m = -1$ and $y = ax^3 + \frac{b}{x}$. The conditions give $a + b = 2$ and $8a + \frac{b}{2} = 8.5$, that is $16a + b = 17$. So $a = 1$ and $b = 1$. At $x = 1.5$: $y = 3.375 + 0.6667 = 4.04$.