Answer:
[tex]y(x)=e^{-3x}[C_1cos(2x)+C_2sin(2x)][/tex]
Step-by-step explanation:
To solve a second-order homogeneous differential equation, we need to substitute each term with the auxiliary equation [tex]am^2+bm+c=0[/tex] where the values of [tex]m[/tex] are the roots:
[tex]y''+6y'+13y=0\\\\m^2+6m+13=0\\\\m^2+6m+13-4=0-4\\\\m^2+6m+9=-4\\\\(m+3)^2=-4\\\\m+3=\pm2i\\\\m=-3\pm2i[/tex]
Since the values of [tex]m[/tex] are complex conjuage roots, then the general solution is [tex]y(x)=e^{\alpha x}[C_1cos(\beta x)+C_2sin(\beta x)][/tex] where [tex]m=\alpha \pm \beta i[/tex].
Thus, the general solution for our given differential equation is [tex]y(x)=e^{-3x}[C_1cos(2x)+C_2sin(2x)][/tex].
