Question

Find particular solution(integral) for the equation \[\large \frac{ d^2 x}{ dt^2 }-9x=9e ^{-12t}\] , worked out k to be \[\pm 3\] , can anyone help me for how to complete?

Answer 1

So the typical solution is: \[
c_1e^{-3t}+c_2e^{3t}
\]

Answer 2

yes that is correct :)

Answer 3

personally I got \[c_{1}e^{3t}+c_{2}e^{-3t}\] but it is the same :)

Answer 4

So you can guess that the particular solution is\[
x_p=Ae^{-12t}
\]Then integrate: \[
x_p'=-12Ae^{-12t}
\]

Answer 5

I mean differentiate

Answer 6

Then do it a second time: \[
x''_p=144Ae^{-12t}
\]

Answer 7

\[x''_{p}=144Ae^{-12t}\] correct

Answer 8

oh you have already done it :P

Answer 9

YEs, now substitute \(x_p\) into the differential equation

Answer 10

\[144Ae^{-12t}-9Ae^{-12t}=9e^{-12t}\] is what I then get is that correct?

Answer 11

Yes.

Answer 12

Notice that\[
144A-9A=9
\]You can solve for \(A\).

Answer 13

so then once solved; \[\large A=\frac {1} {15}\]
and does that make the answer \[\large A=\frac {1}{15} 9e^{-12t}\] ?

Answer 14

\[\large X= \frac {1}{15}9e^{-12t}\] @wio

Answer 15

I think that is the answer, but is there anymore to the question? :)

Answer 16

you had\[x_p=Ae^{-12t}\]


you found A=1/15

so\[x_p=\frac1{15}e^{-12t}\]

Answer 17

right ? @Ray10

Answer 18

Oh of course!! Now I see that :D thank you @UnkleRhaukus

Answer 19

I have another question which I need assistance with, could you have a look at it and see if you can help me please? @UnkleRhaukus

Answer 20

i sure can

Answer 21

I shall post a new question of it :)