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ranyai12
 2 years ago
Help Please!!
Use Laplace transforms to solve the initial value problem
x''+4x'+8x=2e^(t) ; x(0)=0 and x'(0)=4
ranyai12
 2 years ago
Help Please!! Use Laplace transforms to solve the initial value problem x''+4x'+8x=2e^(t) ; x(0)=0 and x'(0)=4

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ranyai12
 2 years ago
Best ResponseYou've already chosen the best response.0this is what I got so far

ranyai12
 2 years ago
Best ResponseYou've already chosen the best response.0after this I am stuck :(

ranyai12
 2 years ago
Best ResponseYou've already chosen the best response.0and this is the table I used: http://tutorial.math.lamar.edu/pdf/Laplace_Table.pdf

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.0use partial fractions on the \[\frac2{s(s+1)(s+4)}\]part and I think it should be pretty straightforward to do inverse Laplace

ranyai12
 2 years ago
Best ResponseYou've already chosen the best response.0but what I do about the 4/s^2+4s

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.0factor out the s and that is what I wrote

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.0oh I see; same thing though factor out s and do partial fractions

ranyai12
 2 years ago
Best ResponseYou've already chosen the best response.0but then you get 4/(s(s+4)>4/s[1/s+4)> 4e^(4t)

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.0\[\frac4{s^2+4s}=\frac As+\frac B{s+4}\implies A(s+4)+Bs=4\]\[A=1~~~,B=1\]

ranyai12
 2 years ago
Best ResponseYou've already chosen the best response.0oh so you do partial fractions with both of them

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.0that's what I'd do, yeah

ranyai12
 2 years ago
Best ResponseYou've already chosen the best response.0and for the other one it's A(s^2+4s) B(s+1) right

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.0\[\mathcal L^{1}\{\frac1s\}=1\]\[\mathcal L^{1}\{\frac1{s+4}\}=e^{4t}\]

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.0for the other one it would be...

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.0\[\frac2{s(s+1)(s+4)}=\frac As+\frac B{s+1}+\frac C{s+4}\]

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.0\[A(s+1)(s+4)+Bs(s+4)+Cs(s+1)=2\]

ranyai12
 2 years ago
Best ResponseYou've already chosen the best response.0then you get As^2+5As+4A+Bs^2+4Bs+Cs^2+C=2

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.0not necessary I don't think plug in s=1,4,and 0 and I think you can get the values quickly

ranyai12
 2 years ago
Best ResponseYou've already chosen the best response.0so you dont need ti distribute it?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.0not necessarily how did I get the other answers? I'll show you...

ranyai12
 2 years ago
Best ResponseYou've already chosen the best response.0i didnt get the answers yet i thiught you had to distribute it then ge tthe answers but now i see that its unnecessary

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.0\[\frac4{s^2+4s}=\frac As+\frac B{s+4}\implies A(s+4)+Bs=4\]now plug in\[s=0:A(0+4)+B(0)=4 \implies 4A=4\implies A=1\]

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.0this is the second term:\[\frac4{s^2+4s}=\frac As+\frac B{s+4}\implies A(s+4)+Bs=4\]now plug in\[s=4:A(4+4)+B(4)=4\implies 4B=4\implies B=1\]

ranyai12
 2 years ago
Best ResponseYou've already chosen the best response.0I didnt get those values though :(

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.0do you see how I got mine?

ranyai12
 2 years ago
Best ResponseYou've already chosen the best response.0i mean for the values of the other one

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.0that I haven't checked yet, one sec...

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.0\[A(s+1)(s+4)+Bs(s+4)+Cs(s+1)=2\]\[s=1:3B=2\implies B=\frac23\]\[s=4:12C=2\implies C=\frac16\]\[s=0:4A=2\implies A=\frac12\]yes I guess you are right :)

ranyai12
 2 years ago
Best ResponseYou've already chosen the best response.0yay!! so would the final answer be (1/2)(2/3)(e^t)+(1/6)e^(4t)e^(4t)

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.0I think you forgot the one from the other term do you have infinite tries?

ranyai12
 2 years ago
Best ResponseYou've already chosen the best response.0no I ownly have 4 more tries

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.0ok let me try it again from the top; I went of what you had I should doublecheck everything

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.0ok I think I found it...

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.0\[x''(0)+4x'+8x=2e^{t}\]\[s^2X(s)sx(0)x'(0)+4sX(s)4x(0)+8X(s)=\frac2{s+1}\]\[(s^2+4s+8)X(s)=\frac2{s+1}+4\]

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.0you dropped the 8...

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.0unfortunately this is going to make partial fractions a real pain...

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.0write is as a complete square\[((s+2)^2+4)X(s)=\frac2{s+1}+4\]you should wind up with some hyperbolic I think

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.0hyperbolic trig functions*

ranyai12
 2 years ago
Best ResponseYou've already chosen the best response.0can you please show me because I honestly have no clue

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.0\[X(s)={2\over{((s+2)^2+4)(s+1)}}+\frac4{((s+2)^2+4)}\]\[={2\over{((s+2)^2+4)(s+1)}}+2\frac2{((s+2)^2+2^2)}\]the second term is form 19 on the chart in your link

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.0\[\mathcal L^{1}\{\frac2{((s+2)^2+2^2)}\}=e^{2t}\sin(2t)\]

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.0the other part... you're gonna have to use partial fractions fractions and it looks like it's gonna be a pain

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.0\[{2\over{((s+2)^2+4)(s+1)}}={As+B\over(s+2)^2+4}+{C\over s+1}\]\[(As+B)(s+1)+C[(s+2)^2+4]=2\]\[s=1:5C=2\implies C=\frac25\]\[s=0:B+\frac{16}5=2\implies B=\frac65\]

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.0I think you'll have to get A on your own ;) I'm a bit busy

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.0I think\[(A+C)s^2=0\implies A=C=\frac25\]but you shouod doublecheck me

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.0I think that is correct
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