GATE 2025 ME – Question 47
Let $y$ be the solution of the differential equation with the initial conditions given below. If $y(x = 2) = A\ln 2$, then the value of $A$ is ____________ (rounded off to 2 decimal places).
$$x^2\frac{d^2y}{dx^2} + 3x\frac{dy}{dx} + y = 0 \quad\quad y(x = 1) = 0 \quad\quad \frac{dy}{dx}(x = 1) = 1$$
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Show answer and explanation
Correct answer: 0.49 to 0.51
Explanation
This is a Cauchy-Euler equation. Put $x = e^t$. It becomes $\frac{d^2y}{dt^2} + 2\frac{dy}{dt} + y = 0$, with the repeated root $-1$, so $y = (C_1 + C_2t)e^{-t} = \frac{C_1 + C_2\ln x}{x}$. From $y(1) = 0$ we get $C_1 = 0$. Then $y = \frac{C_2\ln x}{x}$ and $y'(1) = C_2 = 1$. So $y = \frac{\ln x}{x}$, and $y(2) = \frac{\ln 2}{2}$, which gives $A = 0.50$.