2nd Order Homogenous ODE with complexe roots

Question

Going over past exams (without solutions) I came across this question:

$\frac{d^2y}{dt^2}+2 \frac{dy}{dt}+5y=0$.

I work out using guess $y=\exp\lambda t$

$\lambda^2+2\lambda+5=0$

By quadratic formula: $\lambda =-1 \pm2i$

Therefore $y(t)=\exp(t(-1+2i))$

$= \exp(t)\left( \exp(-1+2i) \right)$

$=\exp(t)\left( \exp(-1(\cos(2)+i\sin(2)\right)))$

$y(t)=e^{t}(\frac{\cos(2)+\sin(2)i}{e^1})$

But when I substitute the found $y$ into the original equation I don't identically get 0, and I'm not sure where I went wrong.

Thanks for your help.

Answer 1

It should be $$\lambda^2+5\lambda+5=0\implies\lambda=\dfrac{-5\pm\sqrt5}2$$

$$\displaystyle y=Ae^{\dfrac{(-5+\sqrt5)t}2}+Be^{\dfrac{(-5-\sqrt5)t}2}$$

EDIT:

After the change in the question,

If $y=e^{-1+2i}$

$$\dfrac{\frac{d^2y}{dt^2}+2 \frac{dy}{dt}+5y}{e^{t(-1+2i})}=(-1+2i)^2+2(-1+2i)+5=-3-4i-2+4i+5=0$$

Similarly for $y=e^{-1-2i}$