Question

Section 5.2 Problem 6:Find the general solution0%3D%200" id="TexFormula1" title="y'' + 6y' + 10y = 0" alt="y'' + 6y' + 10y = 0" align="absmiddle" class="latex-formula">
​

Answer 1

Answer:

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

Step-by-step 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.