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GATE 2021 AE – Question 11

Engineering Mathematics · Differential Equations: First order linear ODEs and higher order linear ODEs with constant coefficients · 1 mark · Multiple choice

The solution of $y^{\prime\prime}+8y^\prime+16y=0$, with $y(0)=1$, $y^\prime(0)=0$, is

  1. $(1+2x)e^{-4x}$
  2. $(1-4x)e^{-4x}$
  3. $(1+8x)e^{-4x}$
  4. $(1+4x)e^{-4x}$

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Correct answer: (D) $(1+4x)e^{-4x}$

Explanation

**Characteristic equation:** $r^2+8r+16=(r+4)^2=0$, so there is a repeated root $r=-4$.

**General solution:**
$$y=(A+Bx)\,e^{-4x}.$$

**Initial conditions.**
- $y(0)=A=1$.
- $y^\prime=Be^{-4x}-4(A+Bx)e^{-4x}$, so $y^\prime(0)=B-4A=0\Rightarrow B=4A=4$.

$$y=(1+4x)\,e^{-4x}\quad(\text{option D}).$$