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Math Help - Differential Equations:Linear first Order Equations

  1. #1
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    Differential Equations:Linear first Order Equations

    How would you solve these? (I use your explanations and answers as reference to my homework, because the summer teacher is out of reach oustide of class).

    1) find an explicit general solution of the differential equation:
    y' - 2xy= e^(x^(2))

    2) Find an explicit general solution of the differential equation:
    xy'= 2y + x^(3) cosx

    3) Find an explicit general and particular solution of the initial value problem:
    xy' - y= x, y(1)= 7

    4) Find an explicit general and particular solution of the initial value problem:
    y' + y= e^(x), y(0)= 1

    5) A 400-gal tank initially contains 100 gal of brine (salt water) containing
    50 lb of salt. Brine containing 1 lb of salt per gallon enters the tank at
    the rate 5 gal/s, and the well-mixed brine in the tank flows out at the rate
    of 3 gal/s. How much salt will the tank contain when it is full of brine?
    (Solve questions using linear first order equation method primarily.
    If you have any other methodic approaches to this problem, please
    feel free to demonstrate).
    Last edited by googoogaga; July 13th 2007 at 12:31 PM.
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  2. #2
    Math Engineering Student
    Krizalid's Avatar
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    Quote Originally Posted by googoogaga View Post
    4) Find an explicit general and particular solution of the initial value problem:
    y' + y= e^(x), y(0)= 1
    Linear ODE y'+P(x)y=Q(x)

    Integrating factor defined by \mu(x)=\exp\int{P(x)}~dx

    In this problem, you can see that the integrating factor is \mu(x)=e^x, then multiplying the equation by this I.F. yields e^xy'+e^xy=e^{2x}\iff(e^xy)'=e^{2x}

    The rest, for you.
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  3. #3
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    5) A 400-gal tank initially contains 100 gal of brine (salt water) containing
    50 lb of salt. Brine containing 1 lb of salt per gallon enters the tank at
    the rate 5 gal/s, and the well-mixed brine in the tank flows out at the rate
    of 3 gal/s. How much salt will the tank contain when it is full of brine?
    (Solve questions using linear first order equation method primarily.
    If you have any other methodic approaches to this problem, please
    feel free to demonstrate).
    At time t there's 100+2t gallons of brine in the tank. So

    \frac{dy}{dt}=\underbrace{5}_{\text{in}}-\overbrace{\frac{3y}{100+2t}}^{\text{out}}

    \frac{dy}{dt}+\frac{3y}{100+2t}=5

    Integrating factor:

    e^{\int\frac{3}{100+2t}dt}=(t+50)^{\frac{3}{2}}

    \frac{d}{dt}((t+50)^{\frac{3}{2}}y)=5(t+50)^{\frac  {3}{2}}

    Integrate and get:

    (t+50)^{\frac{3}{2}}y=2(t+50)^{\frac{5}{2}}+C

    y=2t+100+C(t+50)^{\frac{-3}{2}}

    Using the IC that t(0)=50, we get C=-12500\sqrt{2}

    \boxed{y=2t+100-12500\sqrt{2}(t+50)^{\frac{-3}{2}}}

    The tank is full when 100+2t=400, t=150

    y=2(150)+100-12500\sqrt{2}(150+50)^{\frac{-3}{2}}\approx{393.75}~lbs~of~brine

    Please check my calculations. Easy to go astray.
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  4. #4
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    Problem # 4: Linear First Order Equations( I still did not understand)

    I still did not understand it your explanation. This is what I did to Find the general equation... Is this correct? If not could someone please help me with this?

    y'+ p(x) y= Q(x) p(x)= -2x
    y'- 2xy = e^(x*x) Q(x)= e^(x*x)

    v(x)= e^(Int p(x) dx) = e^ (Int -2x dx)= e^(-x^(2))

    y(x)= Int v(x)Q(x)+c
    v(x)

    y(x)= Int e^(-x^(2)) * e^(x*x) + c
    (e^(-x^(2)
    = e^(-x^(2)) * 1/2 e ^(x*x) +c
    e^(-x^(2))

    G.S. ==> Y(x)= 1/2 e^(x*x) + c
    e^(-x^(2))
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  5. #5
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    Sorry! Wrong Numbering

    I meant problem #1
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  6. #6
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    To Mr. Galactus

    Thanks! That's the Answer I got for the Brine Problem you've helped me with.
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