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Separable First Order Differential Equation

  1. Feb 22, 2012 #1
    1. The problem statement, all variables and given/known data

    [itex] \frac{dy}{dx} = y \sqrt{x} [/itex], f(9) = 5

    3. The attempt at a solution

    [itex] \int dy/y = \int \sqrt{x} dx [/itex]

    [itex] ln |y| = \frac{2}{3} x^\frac{3}{2} + c [/itex]

    [itex] y = e^{\frac{2}{3}x^\frac{3}{2}} + C [/itex]

    [itex] y = Ce^{\frac{2}{3}x^\frac{3}{2}} [/itex]

    [itex] 5 = Ce^{\frac{2}{3}9^\frac{3}{2}} [/itex]

    [itex] 5 = Ce^{18} [/itex]

    [itex] C = \frac{5}{e^{18}} [/itex]

    Thus,[itex] y = \frac{5}{e^{18}} e^{\frac{2}{3}x^\frac{3}{2}} [/itex]
    [itex] y = 5e^{-18} e^{\frac{2}{3}x^\frac{3}{2}} [/itex]
    [itex] y = 5 e^{\frac{2}{3}x^\frac{3}{2}-18} [/itex]
     
    Last edited: Feb 22, 2012
  2. jcsd
  3. Feb 22, 2012 #2
    have ln y + ln c
     
  4. Feb 22, 2012 #3

    SammyS

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    Hello tinopham. Welcome to PF !

    Do you have a question about this ?
     
  5. Feb 22, 2012 #4
    Hi SamS, I was going to ask a question, but I was able to solve it. Thanks!
     
  6. Feb 22, 2012 #5

    HallsofIvy

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    1) The integral of 1/y is ln|y|+ c, not ln y.
    2) Since tinopham had a "c" on the right side ogf the equation, it is not necessary to have a constant on the left. The two constants of integration can be combined on oneside.
     
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