Question

Solve the following IVP:
y'' -4y' +9y=0
y(0)=0
y'(0)=-8
as you can see it is 2nd order homogeneous differential equation any help please?

Answer 1

what is the characteristic polynomial?

Answer 2

r^2-4r+9=0 it is complex root

Answer 3

and what are those roots?

Answer 4

2+root5 i
2- root5 i

Answer 5

characteristic equation is \[D^{2}-4D+9=0\]
ITS SOLUTION IS D=\[(4+\sqrt{20}i)\div2\] & \[(4-\sqrt{20}i)\div2\]
so the solution is

Answer 6

the form is\[y(t)=c_1e^{\lambda t}\cos(\mu t)+c_2\sin(\mu t)\]for complex roots\[\lambda\pm\mu i\]so just plug in the corresponding variables
what do you then get for the general solution?

Answer 7

typo, the general solution is of the form\[y(t)=c_1e^{\lambda t}\cos(\mu t)+c_2e^{\lambda t}\sin(\mu t)\]

Answer 8

ok thanks everyone got it, i just was confused in something now it is all clear thanks

Answer 9

I'm glad :)
welcome!