Solve y+2y+y=0 with y(0)=1 and y(1)=3.

$e^{-x}+[3e\sin(\pi x/2)-1]xe^{-x}$

$e^{-x}-[3e\sin(\pi x/2)-1]xe^{-x}$

GATE 2021 CE (CE1) – Question 36

Engineering Mathematics · ODE: First order and higher order linear equations, Euler-Cauchy equations, initial and boundary value problems · 2 marks · Multiple choice

Solve $y^{\prime\prime}+2y^\prime+y=0$ with $y(0)=1$ and $y(1)=3$.

$e^{-x}+(3e-1)xe^{-x}$
$e^{-x}-(3e-1)xe^{-x}$
$e^{-x}+[3e\sin(\pi x/2)-1]xe^{-x}$
$e^{-x}-[3e\sin(\pi x/2)-1]xe^{-x}$

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Show answer and explanation

Correct answer: (A) $e^{-x}+(3e-1)xe^{-x}$

Explanation

The characteristic equation is $(r+1)^2=0$, so $y=(C_1+C_2x)e^{-x}$. The first condition gives C₁=1; the second gives $(1+C_2)/e=3$, hence C₂=3e−1. This yields **A**. The sine-dependent expressions do not satisfy the differential equation.