# GATE2000: Higher Order Differential Equation

### 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:

(D2 + 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 = eαx (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}