Question

y'' + 4y = {3t, 0 (<=) t (<=)7}, {21, t>7}

for unit step equations you do the laplace transform to both sides, but I can't get the right side unit step function to work properly. What I get is: (u(t) - u(t-7) )3t + 21(u(t-7)) and then you apply the corresponding laplace transform help please?

Answer 1

3t, 0 less than or equal to (t) less than or equal to 7 <- this for piece wise function on the right

Answer 2

\[L\{y''\}=s^2Y(s)-sy(0)-y'(0)\]
right?

Answer 3

right

Answer 4

and Laplace transform of 4y is just 4Y

Answer 5

\[4L\{y(t)\}=4Y(s)\]

Answer 6

then the right side ..... we split up and it messes up our pretty laplace tables :)

Answer 7

i mean i think i set up the unit step functions possibly correct: u(t) -u(t-7) for the first bounds

Answer 8

then u(t-7) for the second bound?

Answer 9

\[\int e^{-st}t\ dt=e^{-st}+\frac{1}{s}\int e^{-st}dt\]
\[\int e^{-st}t\ dt=e^{-st}+\frac{e^{-st}}{s^2};[0,21]\]
\[e^{-21s}+\frac{e^{-21s}}{s^2}-1-\frac{1}{s^2}\]

im not sure what a unit step function is tho

Answer 10

and change 21 to 7 :) this old man cant see too well

Answer 11

hmm but we still havent taken into account that the function = 3t at values of t [0,7]

Answer 12

i forgot a step in my integration by parts; or at least mixed something up on the beginning

and i forgot the 3 .... ugh :/

Answer 13

redo lol

Answer 14

Seems like I am 19 minutes to late.

Answer 15

take your time :D

Answer 16

not sure how to summarize unit step functions, but there a way to characterize the piecewise function without all the inequalities

Answer 17

\[\int 3t\ e^{-st}\ dt\to\ 3\frac{e^{-st}}{-s}+\frac{1}{s}\int e^{-st}dt\]
\[3\frac{e^{-st}}{-s}+\frac{1}{s}(\frac{1}{-s} e^{-st})\]
\[3\frac{e^{-st}}{-s}-\frac{e^{-st}}{s^2}\]

oh theres plenty of room in here

Answer 18

i think that 3 spose to distrribute thru ....

Answer 19

isnt the fact that its 3t only on the interval [0,7] supposed to shift the function over by seven or something?

Answer 20

dunno, havent gotten to shifts yet in class. so far when weve done these we just integrate it on the short interval

Answer 21

hmm i'll try what you have there

Answer 22

when i do it on paper and dont have to focus on the latex coding ... i get:

\[L\{3t\};[0,7]\to\ \frac{-21se^{-7s}-3e^{-7s}+3}{s^2}\]

Answer 23

\[L\{21\}=\frac{21e^{-7s}}{s}\]

Answer 24

how exactly did you do that? O_o