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Thread: Can't integrate this differential equation

  1. #1
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    [SOLVED] Integration of D.E.

    Hi, having difficulty with this D.E. that relates to a tank of initially 400L of pure water, with 1L of salt water (concentration 4g/L) added per min, and 3L of the well-mixed tank solution removed per min. This is what I have so far;

    $\displaystyle Q'(t)+\frac{3Q(t)}{400-2t}=4$

    I'm trying to find an integrating factor though I keep ending up with a final differentiated x(t) function that doesn't make sense (based on t-200, resulting in the function only existing from t > 200 when it should be from t > 0)

    $\displaystyle P(t)=\frac{3}{2(200-t)}$ leading to

    $\displaystyle I(t)=-(200-t)^{3/2}$

    but then this doesn't seem to make the LHS equal to the derivative of $\displaystyle Q(t)I(t)$? Without the negative sign it would work fine but I'm sure that's the integral of P(t)... a little help? Maybe I've done something stupid...


    Last edited by McChickenb; May 4th 2010 at 10:14 AM.
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  2. #2
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    Hello, McChickenb!

    $\displaystyle \frac{dQ}{dt} +\frac{3}{400-2t}\,Q\;=\;4$

    We have: .$\displaystyle \frac{dQ}{dt} + \frac{3}{2}\cdot\frac{1}{200-t}\,Q \;=\;4$


    Integrating factor: .$\displaystyle I \;=\;e^{\frac{3}{2}\!\int \!\frac{dt}{200-t}} \;=\; e^{-\frac{3}{2}\ln(200-t)} \;=\;e^{\ln(200-t)^{-\frac{3}{2}}} \;=\;(200-t)^{-\frac{3}{2}} $


    Multiply by $\displaystyle I\!:\;\;(200-t)^{-\frac{3}{2}}\,\frac{dQ}{dt} + \frac{3}{2}\!\cdot\!\frac{1}{200-t}\!\cdot\!(200-t)^{-\frac{3}{2}} \,Q \;=\;4(200-t)^{-\frac{3}{2}} $


    . . . . . . . . . . . . . . . . . . $\displaystyle (200-t)^{-\frac{3}{2}}+ \frac{3}{2}\,(200-t)^{-\frac{5}{2}}\,Q \;=\;4(200-t)^{-\frac{3}{2}} $


    We have: . $\displaystyle \frac{d}{dt}\bigg[(200-t)^{-\frac{3}{2}}\,Q\bigg] \;=\;4(200-t)^{-\frac{3}{2}} $


    Integrate: . $\displaystyle (200-t)^{-\frac{3}{2}}\,Q \;=\;8(200-t)^{-\frac{1}{2}} + C$


    Multiply by $\displaystyle (200-t)^{\frac{3}{2}}\!:\;\;\;Q \;=\;8(200-t) + C(200-t)^{\frac{3}{2}} $


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  3. #3
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    D'oh! That's perfect. I was confusing myself on the simple stuff. Thanks a lot!
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