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Math Help - Question regard to First Order Differential Equations

  1. #1
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    Question regard to First Order Differential Equations

    Hey guy, I'm having a problem with First order ODE (again)

    The Melting Pot. A tank containing 20,000 litre of blended fuel oil is stirred continuously to keep the mixture homogeneous. The Fuel is blended from a heavy and a light component in equal proportions. The two components are added at a rate of 5 litres/min each, then the blended fuel oil is withdrawn at a rate 10 litres/min to be burnt in a furnace. The blended fuel oil will not burn in the furnace if the heavy component forms more than 80% of the mixture


    At a certain instant, the supply of the light component ceases whilst the heavy component continues to be added at the same rate of 5 litres/meter and the homogeneous mixture is withdrawn at the same rate of 10 litres/min.

    a) let x(t) be the amount of heavy fuel in the tank (in litres) at time t minutes after the supply of light oil ceases. Show that

    \frac{dx}{dt} + \frac{10*x}{20000 - 5*t} = t,  0<=t<4000

    The only problem in question a) I encounter is the "-5*t" can any one tell me where it comes from?



    b) Solve this differential equation for x(t)



    This is my working out for this part

    Question regard to First Order Differential Equations-20131031_181824.jpg


    I got an answer of x(t)=(4000-t)^2*5*(4000-t)

    but the answer in the worksheet is

    x(t)=5*(4000-t) - \frac{1}{1600}*(4000-t)^2

    Can any one tell me where I went wrong?

    Best Regard
    Junks
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  2. #2
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    Re: Question regard to First Order Differential Equations

    If x is the amount of heavy fuel and y is the amount of light fuel at some point in time and these change by \delta x and \delta y in time \delta t, then

    \delta x + \delta y = -5\delta t.

    In the limit as \delta t \rightarrow 0, that gets you \frac{dx}{dt}+\frac{dy}{dt}=-5.

    Integrate and substitute the initial condition (to evaluate the constant of integration) and you find that x+y=20000-5t.

    (Actually, on reflection, you can write this down directly.)

    Now substitute that into the expression that you should have written down for \delta x.

    For your solution to the differential equation, you've forgotten to include a constant of integration.
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