Page 1 of 2 12 LastLast
Results 1 to 15 of 19

Math Help - Question on the applications of Laplace transforms

  1. #1
    Newbie
    Joined
    Oct 2012
    From
    SAUDI ARABI
    Posts
    18

    Question Question on the applications of Laplace transforms

    hi

    I need your help to resolve this question

    y''-y'-2y=18e^-t sin3t

    y(c)=0 , y'(c)=3

    ?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78

    Re: Question on the applications of Laplace transforms

    Quote Originally Posted by loolo View Post
    hi

    I need your help to resolve this question

    y''-y'-2y=18e^-t sin3t

    y(c)=0 , y'(c)=3

    ?
    Recall that the laplace transform of

    \mathcal{L}\{e^{at}f(t)\}=F(s-a)

    So the laplace transform of

    f(t)=18e^{-t}\sin(3t) \to  \frac{54}{(s+1)^2+9}

    Can you finish from here?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Oct 2012
    From
    SAUDI ARABI
    Posts
    18

    Re: Question on the applications of Laplace transforms

    ammmmmm



    I want a solution by this law


    Question on the applications of Laplace transforms-03102012975.jpg
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78

    Re: Question on the applications of Laplace transforms

    Quote Originally Posted by loolo View Post
    ammmmmm



    I want a solution by this law


    Click image for larger version. 

Name:	03102012975.jpg 
Views:	6 
Size:	776.0 KB 
ID:	25057
    I don't see how that rule applies to this problem. Since we don't know the intial contidtions at zero we assume

    y(0)=c_1 \quad y'(0)=c_2

    If we take the laplace transfrom of the equation we get

    s^2Y-sc_1-c_2-sY+c_1-2Y=\frac{54}{(s+1)^2+9}

    Solving for Y gives

    Y=\frac{c_2-c_1}{(s-2)(s+1)}+\frac{c_1s}{(s-2)(s+1)}+\frac{54}{(s-2)(s+1)[(s+1)^2+9]}

    Edit: You could use the theorem to invert the 2nd of the two terms in this problem.
    Last edited by TheEmptySet; October 5th 2012 at 12:35 PM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Oct 2012
    From
    SAUDI ARABI
    Posts
    18

    Re: Question on the applications of Laplace transforms

    i Began resolve the question as follows

    Question on the applications of Laplace transforms-04102012976.jpg
    Last edited by loolo; October 5th 2012 at 01:37 PM.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78

    Re: Question on the applications of Laplace transforms

    Quote Originally Posted by loolo View Post
    i Began resolve the question as follows

    Click image for larger version. 

Name:	04102012976.jpg 
Views:	4 
Size:	1.05 MB 
ID:	25062

    The laplace transform of the product of two function is NOT the product of their transforms.

    To transform the function on the right hand side you must use the s-axis translation theorem. See post #2
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Oct 2012
    From
    SAUDI ARABI
    Posts
    18

    Re: Question on the applications of Laplace transforms

    ammmmmm

    I'll try to find a way s-axis translation theorem.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,487
    Thanks
    1391

    Re: Question on the applications of Laplace transforms

    Are you able to use Partial Fractions to simplify before trying to take the Inverse transform to get back to y? If not, you may need to use the convolution theorem...
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Newbie
    Joined
    Oct 2012
    From
    SAUDI ARABI
    Posts
    18

    Re: Question on the applications of Laplace transforms

    Ammmmmm

    Dr. asked us to use partial fractions
    But I do not know how
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Newbie
    Joined
    Oct 2012
    From
    SAUDI ARABI
    Posts
    18

    Re: Question on the applications of Laplace transforms

    I'm so sorry,

    the Conditions in

    y(0)=0 ,

    y(0)=3




    this is my Try...


    Question on the applications of Laplace transforms-04102012984.jpg
    Last edited by loolo; October 6th 2012 at 02:33 AM.
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Newbie
    Joined
    Oct 2012
    From
    SAUDI ARABI
    Posts
    18

    Re: Question on the applications of Laplace transforms

    Now I want to use partial fractions..

    Can you help me ؟


    Question on the applications of Laplace transforms-04102012985.jpg
    Follow Math Help Forum on Facebook and Google+

  12. #12
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,487
    Thanks
    1391

    Re: Question on the applications of Laplace transforms

    Quote Originally Posted by loolo View Post
    Now I want to use partial fractions..

    Can you help me ؟


    Click image for larger version. 

Name:	04102012985.jpg 
Views:	3 
Size:	792.2 KB 
ID:	25074
    First of all you have an arithmetic error. When you add 3 to the other side and make a common denominator, it should be \displaystyle \begin{align*} \frac{3\left[ (s + 1)^2 + 9 \right]}{(s + 1)^2 + 9} \end{align*}, which when you expand it out gives \displaystyle \begin{align*} \frac{3(s + 1)^2 + 27}{(s + 1)^2 + 9} \end{align*}. As for the rest, you need to solve for \displaystyle \begin{align*} Y(s) \end{align*} then take the inverse transform to get back to \displaystyle \begin{align*} y(t) \end{align*}.
    Follow Math Help Forum on Facebook and Google+

  13. #13
    Newbie
    Joined
    Oct 2012
    From
    SAUDI ARABI
    Posts
    18

    Re: Question on the applications of Laplace transforms

    ammmmmmm

    Can you help me complete the answer
    Follow Math Help Forum on Facebook and Google+

  14. #14
    Senior Member MaxJasper's Avatar
    Joined
    Aug 2012
    From
    Canada
    Posts
    482
    Thanks
    54

    Lightbulb Re: Question on the applications of Laplace transforms

    Quote Originally Posted by loolo View Post
    I need your help to resolve this question
    y''-y'-2y=18e^-t sin3t
    y(c)=0 , y'(c)=3
    In the diff eq, change t to t+c and take LaplaceT and solve for y(t).

    \mathcal{L}_t[y(t)](s)= \frac{3 e^{-c} \left(e^c s^2+2 e^c s+6 s \sin (3 c)+10 e^c+6 \sin (3 c)+18 \cos (3 c)\right)}{\left(s^2-s-2\right) \left(s^2+2 s+10\right)}

    and an inverse LapT would look like:

    y(t)=e^{-3 c-t} \left(e^{c+3 t}+e^{3 t} \sin (3 c)-e^{3 c} \sin (3 t)+e^{3 t} \cos (3 c)+e^{3 c} \cos (3 t)-e^{4 c}-2 e^{3 c} \cos (3 c)\right)
    Last edited by MaxJasper; October 6th 2012 at 06:54 PM.
    Follow Math Help Forum on Facebook and Google+

  15. #15
    Newbie
    Joined
    Oct 2012
    From
    SAUDI ARABI
    Posts
    18

    Re: Question on the applications of Laplace transforms

    Quote Originally Posted by MaxJasper View Post
    In the diff eq, change t to t+c and take LaplaceT and solve for y(t).

    \mathcal{L}_t[y(t)](s)= \frac{3 e^{-c} \left(e^c s^2+2 e^c s+6 s \sin (3 c)+10 e^c+6 \sin (3 c)+18 \cos (3 c)\right)}{\left(s^2-s-2\right) \left(s^2+2 s+10\right)}

    and an inverse LapT would look like:

    y(t)=e^{-3 c-t} \left(e^{c+3 t}+e^{3 t} \sin (3 c)-e^{3 c} \sin (3 t)+e^{3 t} \cos (3 c)+e^{3 c} \cos (3 t)-e^{4 c}-2 e^{3 c} \cos (3 c)\right)


    Quote Originally Posted by loolo View Post
    I'm so sorry,

    the Conditions in

    y(0)=0 ,

    y(0)=3


    Follow Math Help Forum on Facebook and Google+

Page 1 of 2 12 LastLast

Similar Math Help Forum Discussions

  1. Question about Laplace transforms
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: May 8th 2012, 07:13 AM
  2. Laplace transforms
    Posted in the Advanced Applied Math Forum
    Replies: 3
    Last Post: March 23rd 2010, 12:21 AM
  3. The Fourier Transforms and Its Applications
    Posted in the Advanced Applied Math Forum
    Replies: 0
    Last Post: January 22nd 2010, 07:00 PM
  4. Laplace Transforms
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: April 4th 2009, 08:14 AM
  5. laplace transforms
    Posted in the Calculus Forum
    Replies: 5
    Last Post: January 12th 2009, 11:46 PM

Search Tags


/mathhelpforum @mathhelpforum