Question

w''+w=t^2+2 w(0)=1 w'(0)=-1
laplace transforms solving IVP's
some assistance would be nice. we stopped lecture here and didn't get too deep into it so an explanation would be nice

Answer 1

the one example he did like this in class we used y(t)=w(t-1), y'(t)=w'(t-1), y''(t)=w''(t-1)

Answer 2

I haven't read this far in what I was studying x_x But all the stuff I've read on D.Es has been from this site:

http://tutorial.math.lamar.edu/Classes/DE/IVPWithLaplace.aspx

Maybe it'll give some sort of explanation until someone can elaborate further?

Answer 3

cool ill check it out and see

Answer 4

Alright, cool. Hope it helps. I stick around anyway so I can see how it goes, lol.

Answer 5

ok so you take laplace transform of each term
laplace transform of a derivative of any order is :
\[s^{n}F(s)-\sum_{i=1}^{n}s^{(n-i)}f^{(i-1)}(0)\]

Answer 6

so acording to the formula above for laplace transform of w'' we would have
\[s^{2}F(s)-sw(0)-w'(0)\]

Answer 7

can you continue alone ?

Answer 8

i will try giving my daughter a quick bath then ill be back to it...thank you for the help so far i will let you know if i am able to or not

Answer 9

okey if i am here just ask , might not be here bec its quite late might go to sleep

Answer 10

ok

Answer 11

ok back gimme a sec and ill try to work it

Answer 12

not sure :/

Answer 13

could i just substitute the w with y cause after the transformation we are left with the same coefficients

Answer 14

you mean like just change the letter you use like
y''+y=t^2+2

Answer 15

yeah but the only problem i would have would be changing it back into terms of w because i know its not a straight substitution. We didn't even finish the example like this in class we got to partial fractions of the Y(s) he told us to go home and figure it out lol

Answer 16

hm i am not sure but lets solve it without any substitutions bec i am not sure about that

Answer 17

would my substitution be t-0

Answer 18

ok

Answer 19

ok so the laplace of w'' would be s^2W(s)-sw(0)-w'(s)
laplace of w is W(s)
laplace of t^2 is 2/s^3
laplace of 2 is 2/s

so now you need to isolate W(s) in temrs only of s then take partial fractions of the term you get and the take inverse laplace of each fractions and you should get this result :
t^2+cos(t)-sin(t)

Answer 20

my example from class is w''-2w'+w=6t+2 w(-1)=3, w'(-1)=7
then we did
y(t)=w(t-1), y'(t)=w'(t-1), y''(t)=w''(t-1)
converting or original equation to
\[w''(t-1)-2w'(t-1)+w(t-1)=6(t-1)+2\]
which becomes
\[y''-2y'+y=6t-4\]
with initial conditions y(0)=3 and y'(0)=7

Answer 21

ok i will attempt it thank you for the help @litchlani

Answer 22

hm i dunno about that substitution never done it that way ..

Answer 23

that's fine I will work the way you showed and then ask professor. Im just trying to finish all my homework this weekend because i Have two tests on Thursday one for differential equation and the other is physics 2 so my week is gonna be crazy o_O

Answer 24

:\ kinda same here lol having exam in math in 2 weeks also should be crazy a bit for me too xD

Answer 25

then week after next is finals and ill probably have both on the same day too :/ either way that's for the help! have a good night!

Answer 26

thanks* for the help lol

Answer 27

xD you are welcome have nice night aswell ^^