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Differential equation using partial fractions

  1. Apr 13, 2007 #1
    1. The problem statement, all variables and given/known data

    I need to integrate this differential equation using partial fractions to obtain an equation for P in terms of t; P(t):

    1/P dP/dt = b + aP


    2. Relevant equations



    3. The attempt at a solution

    So far, this is what I have:
    ln /P/ = bP + aP^2/2 +c

    Where do I go from here, would I take e from each side and then what would I do? and am I doing this right?
    Please help me
     
    Last edited by a moderator: Apr 13, 2007
  2. jcsd
  3. Apr 13, 2007 #2
    Multiply both sides by P?
     
  4. Apr 13, 2007 #3

    Curious3141

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    Homework Helper

    No sense in doing that, since he still needs to bring the P-terms over to the left hand side.

    Bring the P-terms from the RHS to the LHS, then separate variables :

    [tex]\frac{1}{P(aP+b)}dP = dt[/tex]

    Separate the left hand side using partial fractions (write it as [tex]\frac{k_1}{P} + \frac{k_2}{aP+b}[/tex], then find the values of k1 and k2 quickly with the "Heaviside cover-up" shortcut, for example) and integrate both sides to solve.
     
  5. Apr 13, 2007 #4

    HallsofIvy

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    No. You can't just integrate both sides with respect to P: there is not "dP" on the right side. You can rewrite the equation as
    [tex]\frac{dP}{P(b+aP)}= dt[/tex]
    and integrate- the left side with respect to P, the right side with respect to t.

    As Curious3141 said, you "partial fractions" to integrate the right side.

     
  6. Apr 13, 2007 #5
    partial fractions

    can you point me in the right direction, with all of these variables I keep getting confused?
     
  7. Apr 13, 2007 #6

    HallsofIvy

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    What more "pointing" do you want! Use partial fractions to integrate
    [tex]\frac{dP}{P(b+ aP}[/tex]

    Can you do the "partial fractions" decomposition? You want to find A and B so that
    [tex]\frac{1}{P(b+aP)}= \frac{A}{P}+ \frac{B}{b+ aP}[/tex]
    Multiply both sides of that by P(b+aP) to get
    [tex]1= A(b+aP)+ BP[/tex]
    and solve for A and B. (Hint: let P= 0 and P= -b/a)
     
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