Question

The solution of the differential equation $\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}+y=0$ 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. $e^{\frac{-x}{2}}\{Acos\left(\frac{\sqrt{3}}{2}\right)x+Bsin\left(\frac{\sqrt{3}}{2}\right)x\}$

Answer 1

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

$\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}+y=0$

$D^2+D+1=0$

$D=\frac{-1\pm \sqrt{3}i}{2}$

$y=e^{-x/2}[A\cos \frac{\sqrt{3}}{2}x+B\sin \frac{\sqrt{3}}{2}x]$