I'm stuck with these Laplace problems












3












$begingroup$


I'm stuck on solving (4)(ii) and 4(b), I have tried to solve them but got stuck in the middle, any hints?



enter image description here



This is where I have gotten in (4)(a)(ii) and (4)(b)(i), I have no idea how to start solving on (4)(b)(ii)



enter image description here






calculus laplace-transform







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  • 1




    $begingroup$
    Can you show your work so far and where you got stuck?
    $endgroup$
    – Alex
    Jan 19 at 15:56










  • $begingroup$
    @Alex I have edited and added my work so far
    $endgroup$
    – Youssef Walid
    Jan 19 at 16:07


















3












$begingroup$


I'm stuck on solving (4)(ii) and 4(b), I have tried to solve them but got stuck in the middle, any hints?



enter image description here



This is where I have gotten in (4)(a)(ii) and (4)(b)(i), I have no idea how to start solving on (4)(b)(ii)



enter image description here






calculus laplace-transform







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    Can you show your work so far and where you got stuck?
    $endgroup$
    – Alex
    Jan 19 at 15:56










  • $begingroup$
    @Alex I have edited and added my work so far
    $endgroup$
    – Youssef Walid
    Jan 19 at 16:07
















3












3








3





$begingroup$


I'm stuck on solving (4)(ii) and 4(b), I have tried to solve them but got stuck in the middle, any hints?



enter image description here



This is where I have gotten in (4)(a)(ii) and (4)(b)(i), I have no idea how to start solving on (4)(b)(ii)



enter image description here






calculus laplace-transform







share|cite|improve this question











$endgroup$




I'm stuck on solving (4)(ii) and 4(b), I have tried to solve them but got stuck in the middle, any hints?



enter image description here



This is where I have gotten in (4)(a)(ii) and (4)(b)(i), I have no idea how to start solving on (4)(b)(ii)



enter image description here






calculus laplace-transform




calculus laplace-transform






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 19 at 16:07







Youssef Walid

















asked Jan 19 at 15:53









Youssef WalidYoussef Walid

605




605








  • 1




    $begingroup$
    Can you show your work so far and where you got stuck?
    $endgroup$
    – Alex
    Jan 19 at 15:56










  • $begingroup$
    @Alex I have edited and added my work so far
    $endgroup$
    – Youssef Walid
    Jan 19 at 16:07
















  • 1




    $begingroup$
    Can you show your work so far and where you got stuck?
    $endgroup$
    – Alex
    Jan 19 at 15:56










  • $begingroup$
    @Alex I have edited and added my work so far
    $endgroup$
    – Youssef Walid
    Jan 19 at 16:07










1




1




$begingroup$
Can you show your work so far and where you got stuck?
$endgroup$
– Alex
Jan 19 at 15:56




$begingroup$
Can you show your work so far and where you got stuck?
$endgroup$
– Alex
Jan 19 at 15:56












$begingroup$
@Alex I have edited and added my work so far
$endgroup$
– Youssef Walid
Jan 19 at 16:07






$begingroup$
@Alex I have edited and added my work so far
$endgroup$
– Youssef Walid
Jan 19 at 16:07












1 Answer
1






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0












$begingroup$

For 4aii), Remember your Laplace transform properties and notice that



$$ frac{mathrm{d}}{mathrm{d}s}frac{s}{s^{2}+4} = -frac{s^{2}-4}{(s^{2}+4)^{2}},$$



motivating the manipulation



$$ frac{s^{2}}{s^{2}+4} = frac{1}{2}left(frac{s^{2}-4}{(s^{2}+4)^{2}} + frac{s^{2}+4}{(s^{2}+4)^{2}}right).$$



Alternatively, you can compute the Bromwich integral



$$ f(t) = frac{1}{2pi i}lim_{Rtoinfty}int_{a-iR}^{a+iR}frac{s^{2}}{(s^{2}+4)^{2}},e^{st},mathrm{d}s.$$



It involves finding the residues of the second-order poles at $s = pm 2i$ and summing them.



4bi) is similar. Notice that



$$ frac{9}{(s^{2}+9)^{2}} = frac{1}{2}left(frac{9+s^{2}}{(s^{2}+9)^{2}} + frac{9-s^{2}}{(s^{2}+9)^{2}}right).$$






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    $begingroup$

    For 4aii), Remember your Laplace transform properties and notice that



    $$ frac{mathrm{d}}{mathrm{d}s}frac{s}{s^{2}+4} = -frac{s^{2}-4}{(s^{2}+4)^{2}},$$



    motivating the manipulation



    $$ frac{s^{2}}{s^{2}+4} = frac{1}{2}left(frac{s^{2}-4}{(s^{2}+4)^{2}} + frac{s^{2}+4}{(s^{2}+4)^{2}}right).$$



    Alternatively, you can compute the Bromwich integral



    $$ f(t) = frac{1}{2pi i}lim_{Rtoinfty}int_{a-iR}^{a+iR}frac{s^{2}}{(s^{2}+4)^{2}},e^{st},mathrm{d}s.$$



    It involves finding the residues of the second-order poles at $s = pm 2i$ and summing them.



    4bi) is similar. Notice that



    $$ frac{9}{(s^{2}+9)^{2}} = frac{1}{2}left(frac{9+s^{2}}{(s^{2}+9)^{2}} + frac{9-s^{2}}{(s^{2}+9)^{2}}right).$$






    share|cite|improve this answer









    $endgroup$


















      0












      $begingroup$

      For 4aii), Remember your Laplace transform properties and notice that



      $$ frac{mathrm{d}}{mathrm{d}s}frac{s}{s^{2}+4} = -frac{s^{2}-4}{(s^{2}+4)^{2}},$$



      motivating the manipulation



      $$ frac{s^{2}}{s^{2}+4} = frac{1}{2}left(frac{s^{2}-4}{(s^{2}+4)^{2}} + frac{s^{2}+4}{(s^{2}+4)^{2}}right).$$



      Alternatively, you can compute the Bromwich integral



      $$ f(t) = frac{1}{2pi i}lim_{Rtoinfty}int_{a-iR}^{a+iR}frac{s^{2}}{(s^{2}+4)^{2}},e^{st},mathrm{d}s.$$



      It involves finding the residues of the second-order poles at $s = pm 2i$ and summing them.



      4bi) is similar. Notice that



      $$ frac{9}{(s^{2}+9)^{2}} = frac{1}{2}left(frac{9+s^{2}}{(s^{2}+9)^{2}} + frac{9-s^{2}}{(s^{2}+9)^{2}}right).$$






      share|cite|improve this answer









      $endgroup$
















        0












        0








        0





        $begingroup$

        For 4aii), Remember your Laplace transform properties and notice that



        $$ frac{mathrm{d}}{mathrm{d}s}frac{s}{s^{2}+4} = -frac{s^{2}-4}{(s^{2}+4)^{2}},$$



        motivating the manipulation



        $$ frac{s^{2}}{s^{2}+4} = frac{1}{2}left(frac{s^{2}-4}{(s^{2}+4)^{2}} + frac{s^{2}+4}{(s^{2}+4)^{2}}right).$$



        Alternatively, you can compute the Bromwich integral



        $$ f(t) = frac{1}{2pi i}lim_{Rtoinfty}int_{a-iR}^{a+iR}frac{s^{2}}{(s^{2}+4)^{2}},e^{st},mathrm{d}s.$$



        It involves finding the residues of the second-order poles at $s = pm 2i$ and summing them.



        4bi) is similar. Notice that



        $$ frac{9}{(s^{2}+9)^{2}} = frac{1}{2}left(frac{9+s^{2}}{(s^{2}+9)^{2}} + frac{9-s^{2}}{(s^{2}+9)^{2}}right).$$






        share|cite|improve this answer









        $endgroup$



        For 4aii), Remember your Laplace transform properties and notice that



        $$ frac{mathrm{d}}{mathrm{d}s}frac{s}{s^{2}+4} = -frac{s^{2}-4}{(s^{2}+4)^{2}},$$



        motivating the manipulation



        $$ frac{s^{2}}{s^{2}+4} = frac{1}{2}left(frac{s^{2}-4}{(s^{2}+4)^{2}} + frac{s^{2}+4}{(s^{2}+4)^{2}}right).$$



        Alternatively, you can compute the Bromwich integral



        $$ f(t) = frac{1}{2pi i}lim_{Rtoinfty}int_{a-iR}^{a+iR}frac{s^{2}}{(s^{2}+4)^{2}},e^{st},mathrm{d}s.$$



        It involves finding the residues of the second-order poles at $s = pm 2i$ and summing them.



        4bi) is similar. Notice that



        $$ frac{9}{(s^{2}+9)^{2}} = frac{1}{2}left(frac{9+s^{2}}{(s^{2}+9)^{2}} + frac{9-s^{2}}{(s^{2}+9)^{2}}right).$$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 20 at 3:03









        IninterrompueIninterrompue

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