Find the solution for the spring-mass problem $y′′+9y=cos(3t)$. Solve with initial conditions $y(0) = 0$, $y′ (0) = 0$. Using Laplace transform
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I first took the Laplace transform of each part then getting $s^2Y+9Y=fracss^2+9$ then solving for Y, I got $Y=fracs(s^2+9)^2$ but don't know how to simplify that to something that can be transformed back.
differential-equations laplace-transform
asked May 3 '15 at 2:31
Zoe Hayes
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I first took the Laplace transform of each part then getting $s^2Y+9Y=fracss^2+9$ then solving for Y, I got $Y=fracs(s^2+9)^2$ but don't know how to simplify that to something that can be transformed back.
differential-equations laplace-transform
asked May 3 '15 at 2:31
Zoe Hayes
1056
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I first took the Laplace transform of each part then getting $s^2Y+9Y=fracss^2+9$ then solving for Y, I got $Y=fracs(s^2+9)^2$ but don't know how to simplify that to something that can be transformed back.
differential-equations laplace-transform
asked May 3 '15 at 2:31
Zoe Hayes
1056
I first took the Laplace transform of each part then getting $s^2Y+9Y=fracss^2+9$ then solving for Y, I got $Y=fracs(s^2+9)^2$ but don't know how to simplify that to something that can be transformed back.
differential-equations laplace-transform
asked May 3 '15 at 2:31
Zoe Hayes
1056
edited May 3 '15 at 6:52
user236002
asked May 3 '15 at 2:31
Zoe Hayes
1056
asked May 3 '15 at 2:31
Zoe Hayes
1056
asked May 3 '15 at 2:31
Zoe Hayes
1056
1056
add a comment |Â
add a comment |Â
4 Answers
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Hint: $fracs(s^2+9)^2 = fracddsfrac-12(s^2+9)$
answered May 3 '15 at 2:34
Mehdi Jafarnia Jahromi
1,134515
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With convolution
$$dfracs(s^2+9)^2=dfrac13dfracss^2+9dfrac3s^2+9=dfrac13int_0^tcos 3xsin3(t-x) dx=dfrac16tsin3t$$
answered Aug 29 at 5:13
Nosrati
21.9k51747
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-1
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If you let k=3, then you can use that fact that L.T. t*sin(kt) = 2ks/(s^2+k^2)^2 or in your case, 2*3*s/(s^2+3^2)^2 -> 6s/(s^2+3^2)^2 which if you invert should lead to (1/6)t*sint(3t). Try that.
answered May 3 '15 at 2:39
JohnWayne360
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$$mathcalLt sin kt = - fracddsmathcalLsin kt =
- fracddsleft( fracks^2+k^2 right) = frac2ks(s^2 + k^2)^2$$
edited Aug 29 at 6:47
Andrew
2,38221223
answered Aug 29 at 5:01
Daniel Méndez
1
Please, avoid link-only answers. Type an explanation.
– Taroccoesbrocco
Aug 29 at 5:06
Hello there. For various kinds of reasons, the answer with a link only may not be accepted, since some users might not be able to see it. Take a few minutes to learn MathJax and type your explanation.
– xbh
Aug 29 at 6:06
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4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Hint: $fracs(s^2+9)^2 = fracddsfrac-12(s^2+9)$
answered May 3 '15 at 2:34
Mehdi Jafarnia Jahromi
1,134515
add a comment |Â
up vote
1
down vote
Hint: $fracs(s^2+9)^2 = fracddsfrac-12(s^2+9)$
answered May 3 '15 at 2:34
Mehdi Jafarnia Jahromi
1,134515
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Hint: $fracs(s^2+9)^2 = fracddsfrac-12(s^2+9)$
answered May 3 '15 at 2:34
Mehdi Jafarnia Jahromi
1,134515
Hint: $fracs(s^2+9)^2 = fracddsfrac-12(s^2+9)$
answered May 3 '15 at 2:34
Mehdi Jafarnia Jahromi
1,134515
answered May 3 '15 at 2:34
Mehdi Jafarnia Jahromi
1,134515
answered May 3 '15 at 2:34
Mehdi Jafarnia Jahromi
1,134515
answered May 3 '15 at 2:34
Mehdi Jafarnia Jahromi
1,134515
1,134515
add a comment |Â
add a comment |Â
up vote
0
down vote
With convolution
$$dfracs(s^2+9)^2=dfrac13dfracss^2+9dfrac3s^2+9=dfrac13int_0^tcos 3xsin3(t-x) dx=dfrac16tsin3t$$
answered Aug 29 at 5:13
Nosrati
21.9k51747
add a comment |Â
up vote
0
down vote
With convolution
$$dfracs(s^2+9)^2=dfrac13dfracss^2+9dfrac3s^2+9=dfrac13int_0^tcos 3xsin3(t-x) dx=dfrac16tsin3t$$
answered Aug 29 at 5:13
Nosrati
21.9k51747
add a comment |Â
up vote
0
down vote
up vote
0
down vote
With convolution
$$dfracs(s^2+9)^2=dfrac13dfracss^2+9dfrac3s^2+9=dfrac13int_0^tcos 3xsin3(t-x) dx=dfrac16tsin3t$$
answered Aug 29 at 5:13
Nosrati
21.9k51747
With convolution
$$dfracs(s^2+9)^2=dfrac13dfracss^2+9dfrac3s^2+9=dfrac13int_0^tcos 3xsin3(t-x) dx=dfrac16tsin3t$$
answered Aug 29 at 5:13
Nosrati
21.9k51747
answered Aug 29 at 5:13
Nosrati
21.9k51747
answered Aug 29 at 5:13
Nosrati
21.9k51747
answered Aug 29 at 5:13
Nosrati
21.9k51747
21.9k51747
add a comment |Â
add a comment |Â
up vote
-1
down vote
If you let k=3, then you can use that fact that L.T. t*sin(kt) = 2ks/(s^2+k^2)^2 or in your case, 2*3*s/(s^2+3^2)^2 -> 6s/(s^2+3^2)^2 which if you invert should lead to (1/6)t*sint(3t). Try that.
answered May 3 '15 at 2:39
JohnWayne360
1458
add a comment |Â
up vote
-1
down vote
If you let k=3, then you can use that fact that L.T. t*sin(kt) = 2ks/(s^2+k^2)^2 or in your case, 2*3*s/(s^2+3^2)^2 -> 6s/(s^2+3^2)^2 which if you invert should lead to (1/6)t*sint(3t). Try that.
answered May 3 '15 at 2:39
JohnWayne360
1458
add a comment |Â
up vote
-1
down vote
up vote
-1
down vote
If you let k=3, then you can use that fact that L.T. t*sin(kt) = 2ks/(s^2+k^2)^2 or in your case, 2*3*s/(s^2+3^2)^2 -> 6s/(s^2+3^2)^2 which if you invert should lead to (1/6)t*sint(3t). Try that.
answered May 3 '15 at 2:39
JohnWayne360
1458
If you let k=3, then you can use that fact that L.T. t*sin(kt) = 2ks/(s^2+k^2)^2 or in your case, 2*3*s/(s^2+3^2)^2 -> 6s/(s^2+3^2)^2 which if you invert should lead to (1/6)t*sint(3t). Try that.
answered May 3 '15 at 2:39
JohnWayne360
1458
answered May 3 '15 at 2:39
JohnWayne360
1458
answered May 3 '15 at 2:39
JohnWayne360
1458
answered May 3 '15 at 2:39
JohnWayne360
1458
1458
add a comment |Â
add a comment |Â
up vote
-1
down vote
$$mathcalLt sin kt = - fracddsmathcalLsin kt =
- fracddsleft( fracks^2+k^2 right) = frac2ks(s^2 + k^2)^2$$
edited Aug 29 at 6:47
Andrew
2,38221223
answered Aug 29 at 5:01
Daniel Méndez
1
Please, avoid link-only answers. Type an explanation.
– Taroccoesbrocco
Aug 29 at 5:06
Hello there. For various kinds of reasons, the answer with a link only may not be accepted, since some users might not be able to see it. Take a few minutes to learn MathJax and type your explanation.
– xbh
Aug 29 at 6:06
add a comment |Â
up vote
-1
down vote
$$mathcalLt sin kt = - fracddsmathcalLsin kt =
- fracddsleft( fracks^2+k^2 right) = frac2ks(s^2 + k^2)^2$$
edited Aug 29 at 6:47
Andrew
2,38221223
answered Aug 29 at 5:01
Daniel Méndez
1
Please, avoid link-only answers. Type an explanation.
– Taroccoesbrocco
Aug 29 at 5:06
Hello there. For various kinds of reasons, the answer with a link only may not be accepted, since some users might not be able to see it. Take a few minutes to learn MathJax and type your explanation.
– xbh
Aug 29 at 6:06
add a comment |Â
up vote
-1
down vote
up vote
-1
down vote
$$mathcalLt sin kt = - fracddsmathcalLsin kt =
- fracddsleft( fracks^2+k^2 right) = frac2ks(s^2 + k^2)^2$$
edited Aug 29 at 6:47
Andrew
2,38221223
answered Aug 29 at 5:01
Daniel Méndez
1
$$mathcalLt sin kt = - fracddsmathcalLsin kt =
- fracddsleft( fracks^2+k^2 right) = frac2ks(s^2 + k^2)^2$$
edited Aug 29 at 6:47
Andrew
2,38221223
answered Aug 29 at 5:01
Daniel Méndez
1
edited Aug 29 at 6:47
Andrew
2,38221223
edited Aug 29 at 6:47
Andrew
2,38221223
edited Aug 29 at 6:47
Andrew
2,38221223
2,38221223
answered Aug 29 at 5:01
Daniel Méndez
1
answered Aug 29 at 5:01
Daniel Méndez
1
answered Aug 29 at 5:01
Daniel Méndez
1
1
Please, avoid link-only answers. Type an explanation.
– Taroccoesbrocco
Aug 29 at 5:06
Hello there. For various kinds of reasons, the answer with a link only may not be accepted, since some users might not be able to see it. Take a few minutes to learn MathJax and type your explanation.
– xbh
Aug 29 at 6:06
add a comment |Â
Please, avoid link-only answers. Type an explanation.
– Taroccoesbrocco
Aug 29 at 5:06
Hello there. For various kinds of reasons, the answer with a link only may not be accepted, since some users might not be able to see it. Take a few minutes to learn MathJax and type your explanation.
– xbh
Aug 29 at 6:06
Please, avoid link-only answers. Type an explanation.
– Taroccoesbrocco
Aug 29 at 5:06
Please, avoid link-only answers. Type an explanation.
– Taroccoesbrocco
Aug 29 at 5:06
Hello there. For various kinds of reasons, the answer with a link only may not be accepted, since some users might not be able to see it. Take a few minutes to learn MathJax and type your explanation.
– xbh
Aug 29 at 6:06
Hello there. For various kinds of reasons, the answer with a link only may not be accepted, since some users might not be able to see it. Take a few minutes to learn MathJax and type your explanation.
– xbh
Aug 29 at 6:06
add a comment |Â
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