Find the general solution of the differential equation
𝑦″+2𝑦′+2𝑦=0
The characteristic equation of the given equation is:
$𝑟^{2}$+2𝑟+2=0
𝑟=$\frac{−𝑏±\sqrt{𝑏^{2}−4𝑎𝑐}}{2𝑎}$=$\frac{−2±\sqrt{-4}}{2}$=−1±𝑖
Then,
$𝑟_{1}$=−1+𝑖 $𝑟_{2}$=−1−𝑖
Therefore, the general solution of the given differential equation is:
𝑦=$𝑐_{1}𝑒^{−𝑡𝑐𝑜𝑠𝑡}+𝑐_{2}𝑒^{−𝑡𝑠𝑖𝑛𝑡}$
