Results 1 to 6 of 6
Like Tree2Thanks
  • 1 Post By BobP
  • 1 Post By HallsofIvy

Math Help - Partial fractions question

  1. #1
    Member
    Joined
    Apr 2010
    Posts
    116
    Thanks
    1

    Partial fractions question

    Hello, I am in a differential equations class need to find the inverse Laplace of
    \frac{1}{s(s^2+5)}, but have forgotten how to do partial fractions with quadratic factors. What I got so far:

    \frac{1}{s(s^2+5)}=\frac{A}{s}+\frac{Bs+C}{s^2+5}

    A(s^2+5)+(Bs+C)s=1

    Letting s=0 I get A=\frac{1}{5}, but where do I go from there?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member MaxJasper's Avatar
    Joined
    Aug 2012
    From
    Canada
    Posts
    482
    Thanks
    54

    Lightbulb Re: Partial fractions question

    \frac{1}{s\left(s^2+5\right)} =    \frac{1}{5 s}-\frac{s}{5 \left(5+s^2\right)}
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Apr 2010
    Posts
    116
    Thanks
    1

    Re: Partial fractions question

    Thanks! Could you explain how you got that?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Jun 2009
    Posts
    660
    Thanks
    133

    Re: Partial fractions question

    Equate coefficients of the various powers of s across the identity.

    The number of s^{2} on the LHS have to equal the number of s^{2} on the RHS.

    That gives you A+B=0.

    Then equate coefficients of s to get the value of C.

    Alternatively you could substitute two other values for s and solve the resulting simultaneous equations.
    Thanks from Ragnarok
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,701
    Thanks
    1470

    Re: Partial fractions question

    You have A(s^2+ 5)+ (Bs+ C)s= 1 which is to be true for all s. So take s to be some simple numbers.

    If s= 0, then A(0+ 5)+ (B(0)+ C)0= 5A= 1. Putting that value into the equation, (1/5)(s^2+ 5)+ (Bs+ C)s= (1/5+ B)s^2+ Cs+ 1= 1 or (1/5+ B)s^2+ Cs= 0.
    Taking s= 1, that becomes 1/5+ B+ C= 0 or B+ C= -1/5. Taking s= -1, it is 1/5+ B- C= 0 or B- C= 1/5. Adding those, 2B= 0 so B= 0. Subtracting, 2C= -2/5 so C= -1/5.

    Another way to get that is to multiply out A(s^2+ 5)+ (Bs+ C)s=(A+ B)s^2+ Cs+ 5A= 0s^2+ 0s+ 1. Now, because that is true for all s, we must have "corresponding coefficients" equal: A+ B= 0, C= 0, 5A= 1. That gives A= 1/5 and then (1/5)+ B= 0 so B= -1/5.
    Thanks from Ragnarok
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Member
    Joined
    Apr 2010
    Posts
    116
    Thanks
    1

    Re: Partial fractions question

    Aah! Thanks a lot!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 12
    Last Post: October 2nd 2011, 06:07 AM
  2. partial fractions question
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: September 14th 2010, 11:47 AM
  3. Quick Question on Partial Fractions Problem
    Posted in the Algebra Forum
    Replies: 3
    Last Post: March 26th 2010, 09:49 PM
  4. Partial Fractions Question
    Posted in the Algebra Forum
    Replies: 3
    Last Post: March 26th 2010, 04:24 AM
  5. Partial Fractions Question
    Posted in the Calculus Forum
    Replies: 4
    Last Post: January 4th 2010, 04:49 PM

Search Tags


/mathhelpforum @mathhelpforum