### Question: GATE2008 (2 Marks)

It is given that \(y” + 2y’+ y = 0, y(0) = 0, y(1) = 0 \). What is y(0.5) $

(A) 0 | (B) 0.37 |

(C) 0.62 | (D) 1.13 |

### Solution:

(D^{2} + 2D + 1) y = 0

(D + 1)^{2} y = 0

D = –1, –1 (real repeated roots; similar to (D – a)^{2} y = 0; having solution y = (C_{1} + C_{2}x)e^{ax}

In this case, a = –1

Solution y = (C_{1} + C_{2}x)e^{–x}

Applying boundary conditions:

y(0) = 0

0 = C_{1}

y(1) = 0

0 = C_{1} + C_{2}

C_{2} = 0

Equation becomes y = 0

Therefore, y (0.5) = 0