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Solving a Differential Equation

  1. May 4, 2010 #1
    1. The problem statement, all variables and given/known data
    [tex]
    \[P^{'}(t)+(\lambda +\mu )P(t)=\lambda \]

    [/tex]

    I have never worked with differential equations before and I am trying to work off of the one example we did in class, but I can't figure out where I am going wrong.

    2. Relevant equations



    3. The attempt at a solution
    The first thing I did was multiply both sides by
    [tex]
    \[e^{(\lambda +\mu )t}\]
    [/tex]

    Then,
    [tex]
    \[\frac{d}{dt}[e^{(\lambda + \mu)t}P(t)]=\lambda e^{(\lambda + \mu)t}\]
    [/tex]

    Integrating both sides,
    [tex]
    \[e^{(\lambda + \mu)t}P(t)=\frac{\lambda e^{(\lambda + \mu)t}}{\lambda + \mu} + C\]
    [/tex]

    which seems to give me
    [tex]
    \[P(t)=\frac{\lambda}{\lambda + \mu}\]
    [/tex]

    but I know that this is not correct since I am supposed to showing that the solution is
    [tex]
    \[P(t)=\frac{\lambda}{\lambda + \mu}(1 - e^{-(\lambda + \mu)t})+P(0)e^{-(\lambda + \mu)t}\]
    [/tex].

    I don't think I am solving for C correctly but since I have never really been taught this I'm not quite sure what to do or how to get that solution. I'd really appreciate it if someone could let me know where I am going wrong.
     
  2. jcsd
  3. May 4, 2010 #2

    Mark44

    Staff: Mentor

    You were doing great up to here (above). You multiplied both sides of the equation by
    [tex]
    e^{-(\lambda + \mu)t}
    [/tex]
    but forgot to multiply the constant C.
     
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