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GATE 2022 AE – Question 37

Engineering Mathematics · Differential Equations: First order linear ODEs and higher order linear ODEs with constant coefficients · 2 marks · Multiple choice

For $y\prime\prime-2y\prime+y=0$, y(0)=0 and y′(0)=1, the value y(1/2) is

  1. 0
  2. $\sqrt e$
  3. $\sqrt e/2$
  4. $\sqrt{e/2}$

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Correct answer: (C) $\sqrt e/2$

Explanation

**Characteristic equation:** $y^{\prime\prime}-2y^\prime+y=0\Rightarrow r^2-2r+1=(r-1)^2=0$, a repeated root $r=1$.

**General solution:**
$$y=(C_1+C_2x)\,e^{x}.$$

**Initial conditions.**
- $y(0)=C_1=0$.
- $y^\prime=C_2e^x+(C_1+C_2x)e^x$, so $y^\prime(0)=C_2+C_1=C_2=1$.

So $y=xe^x$.

**Value at $x=\tfrac12$:**
$$y\left(\tfrac12\right)=\tfrac12e^{1/2}=\frac{\sqrt e}{2}\quad(\text{option C}).$$