### Question: GATE2000 (1 Mark)

The solution of the differential equation \(\displaystyle \frac{d^2y}{dx^2}+ \frac{dy}{dx} + y = 0 \)

\((A) Ae^x + Be^{-x} \) |

\((B) e^x (Ax + B) \) |

\((C) e^{-x} {{A cos(\sqrt{3}/2) x + B cos(\sqrt{3}/2)x}} \) |

\((D) e^{-x/2} {{A cos (\sqrt{3}/2) x +B cos(\sqrt{3}/2) x}} \) |

### Solution:

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

\(\displaystyle D =\frac{-1 \pm \sqrt{1 – 4}}{2} = \frac{-1 \pm \sqrt{3}i}{2} \\ \displaystyle = – \frac{1}{2} \pm \frac{\sqrt{3}}{2}i \)

This is of the form (D – a)(D – b)y = 0;

Where a = α + iβ and b = α – iβ which has solution y = e^{αx} (Acosβx + Bsinβx)

In this case, α = \(\displaystyle – \frac{1}{2}\) and β = \(\displaystyle \frac{\sqrt{3}}{2} i \)

Solution: y = e^{-x/2}{Acos(\(\sqrt{3}\) /2)x + Bsin(\(\sqrt{3} \) /2)x}