Question

The solution of the diffeential equaion $\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}+y=0$ is

• A. $A{e}^x+B{e}^{-x}$
• B. $e^x(Ax+B)$
• C. $e^{-x}\{Acos\left(\frac{\sqrt{3}}{2}\right)x+Bcos\left(\frac{\sqrt{3}}{2}\right)x\}$
• D. $e^{\frac{-x}{2}}\{Acos\left(\frac{\sqrt{3}}{2}\right)x+Bcos\left(\frac{\sqrt{3}}{2}\right)x\}$

Answer 1

The correct option is D $e^{\frac{-x}{2}}\{Acos\left(\frac{\sqrt{3}}{2}\right)x+Bcos\left(\frac{\sqrt{3}}{2}\right)x\}$

$\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}+y=0$
$D^2$ + D + 1 = 0
Auxiliary equation is
$m^2$ + m + 1 = 0
m = $\frac{-1\pm \sqrt{3}i}{2}$
General solution is
$y=e^{\frac{-x}{2}}\{Acos\left(\frac{\sqrt{3}}{2}\right)x+Bcos\left(\frac{\sqrt{3}}{2}\right)x\}$