Question

Use Laplace transforms to solve the initial value problem

x''+4x'+8x=2e^(−t) x(0)=0 x'(0)=4

Answer 1

@richyw

Answer 2

are you allowed to use a table?

Answer 3

this is my work and im not sure what i did wrong

Answer 4

and yes i am

Answer 5

i'm already confused lol. where did you get the 18 from?

Answer 6

ooops thats supposed to be an 8

Answer 7

yeah if you put an 8 it looks pretty straightforward as you clearly know what to do. I never finished it though as I am cramming for calculus haha!

Answer 8

id you get x=(4s+6)/(s=1)(s^2+4a+8)

Answer 9

that part always confuses me so if you can please check that id be fine

Answer 10

(s-1)*

Answer 11

yes I got there that's where I quit. I'm thinking about it again now

Answer 12

ok thanks

Answer 13

hmm after my partial fractions I'm getting one that is hard to fix into a nice way to inverse laplace

Answer 14

what did you end up getting?

Answer 15

I can't figure out the answer. I really do apologize but I have a big final coming up and rushed through it. I don't mean to leave you hanging!

Answer 16

its ok thanks i just wanted what x= to anyway not the whole thing but I totally understand!!! Finals suck!

Answer 17

@jagan

Answer 18

x'' + 4x' +8x= 2*exp(-t), x(0)=0, x'(0)= 4

\[ (s^2+4s+8)L[x] -(s+4)*0 - (1)*4 = \frac{2}{s+1} \]
\[ (s^2+4s+8)L[x] = \frac{2}{s+1} +4 \]
\[ L[x]= \frac{4s+6}{(s+1)(s^2+4s+8)} \]

partial fraction expansion
\[ L[x]= \frac{2}{5}\frac{1}{(s+1)} -\frac{2}{5}\frac{s-7}{s^2+4s+8} \]

complete the square on the denominator of the 2nd term to match the laplace tables
rewrite the numerator
\[ L[x]= \frac{2}{5}\frac{1}{(s+1)} -\frac{2}{5}\frac{s+2-9}{(s+2)^2+2^2} \]
re-write the last term as 2 fractions that match the Laplace tables.
\[ L[x]= \frac{2}{5}\frac{1}{(s+1)} -\frac{2}{5}\frac{s+2}{(s+2)^2+2^2}+\frac{9}{5}\frac{2}{(s+2)^2+2^2} \]
\[ x= \frac{2}{5}e^{-t}- \frac{2}{5}e^{-2t}cos(2t)+\frac{9}{5}e^{-2t}sin(2t) \]

see http://en.wikipedia.org/wiki/Laplace_transform#Table_of_selected_Laplace_transforms

Answer 19

I just gave the highlights, as this takes quite a bit of work. Ask if you have any questions.