Section 5.2 Problem 6: Find the general solution y'' + 6y' + 10y = 0 ​

Question

Section 5.2 Problem 6:

Find the general solution

`y'' + 6y' + 10y = 0`
​

Answer 1

`y=e^(-3t)(A\: cos\: t+B\:sin\:t)`

Explanation:

Given Second-Order Homogenous Differential Equation

`y''+6y'+10y=0`

Use Auxiliary Equation

`m^2+6m+10=0\\\\(m+3)^2+1=0\\\\(m+3)^2=-1\\\\m+3=\pm i\\\\m=-3\pm i`

General Solution

`y=e^(pt)(A\: cos\: qt+B\:sin\:qt)\\\\y=e^(-3t)(A\: cos\: t+B\:sin\:t)`

Note that the DE has two distinct complex solutions
`p\pm qi` where
`A` and
`B` are arbitrary constants.