Question

Solve the ordinary differential equation
d2y/dx2+dy/dx+y=0

Answer 1

$y''+y'+y=0\\ \lambda^2+\lambda+1=0\\ D=1-4=-3=3i^2\\ \lambda_1=\frac{-1-i\sqrt3}{2}\\ \lambda_2=\frac{-1+i\sqrt3}{2}\\$

Fundamental system is

$y_1=e^{-\frac{1}{2}x}\cos\frac{\sqrt3}{2}x\\ y_2=e^{-\frac{1}{2}x}\sin\frac{\sqrt3}{2}x\\$

Solution of equation is

$y=c_1e^{-\frac{1}{2}x}\cos\frac{\sqrt3}{2}x+\\ +c_2e^{-\frac{1}{2}x}\sin\frac{\sqrt3}{2}x$