GATE AE 2022 • Q37 MCQ • 2 Marks Engineering Mathematics

Engineering Mathematics

Differential Equations

Subtopic: First Order ODEs

Question Statement:

Consider the differential equation $\frac{d^2 y}{dx^2} - 2 \frac{dy}{dx} + y = 0$. The boundary conditions are $y = 0$ and $\frac{dy}{dx} = 1$ at $x = 0$. Then the value of $y$ at $x = 1/2$ is:
Question figure for GATE AE 2022 Q37

Scientific schematic diagram reproduced from official IIT paper

Options:

  • (A) 0
  • (B) $\sqrt{e}$
  • (C) $\frac{\sqrt{e}}{2}$
  • (D) $\frac{e}{2}$

Official Answer Key:

C

Step-by-Step Solution & Explanation

### **Detailed Step-by-Step Solution (GATE AE 2022 Q.37)** **1. Characteristic Equation**: $$m^2 - 2m + 1 = 0 \implies (m - 1)^2 = 0 \implies m = 1 \text{ (repeated root)}$$ General solution: $$y(x) = (C_1 + C_2 x) e^x$$ **2. Applying Initial Conditions**: - $y(0) = 0 \implies C_1 = 0 \implies y(x) = C_2 x e^x$ - $y'(x) = C_2 (e^x + x e^x)$ - $y'(0) = 1 \implies C_2(1 + 0) = 1 \implies C_2 = 1$ $$y(x) = x e^x$$ **3. Value at $x = 1/2$**: $$y(1/2) = \frac{1}{2} e^{1/2} = \frac{\sqrt{e}}{2}$$ - **Official Answer Key**: **C**
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