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Solving an ODE

  1. Jan 27, 2006 #1
    Hi all!
    How to solve this ODE?
    [tex] \frac{a+b-c\sqrt{H}}{k}=\frac{dH}{dt}[/tex] , where a,b, c and k are constants; H is the variable

    I am up to this step:
    [tex]\int \frac{dH}{a+b-c\sqrt{H}}=\int \frac{dt}{k}[/tex]
    and I don't know how to integrate the left integral, can anyone help please?
     
    Last edited: Jan 27, 2006
  2. jcsd
  3. Jan 27, 2006 #2
    Hi,


    [tex] I = \int \frac { dH} { a + b- c \sqrt H } [/tex]

    Let u2= H >>> 2u du = dH

    [tex] \therefore I = \int \frac {2u } { a + b- c u } du
    = \frac {-2} {c} \int \frac { -cu + a + b- (a + b) } { a + b- cu } du
    = = \frac {-2} {c} \left( 1 - \frac { a + b} { -c } \int \frac { -c } { a + b- cu } du \right)[/tex]

    [tex] = \frac { -2 } {c} u - \frac { 2(a + b) } {c^2} \ln | a + b- c u | + C
    = \frac { -2 } {c} \sqrt {H} - \frac { 2(a + b) } {c^2} \ln | a + b- c \sqrt {H} | + C[/tex]
     
    Last edited: Jan 27, 2006
  4. Jan 27, 2006 #3

    Many thanks.
    But it is very hard to have a closed form of H in terms of t for the solution then?
     
  5. Jan 28, 2006 #4
    It is not necessary to have an explicit relation, implicit relation is sufficuint
     
  6. Jan 28, 2006 #5
    oops..sorry..
    I have made a mistake in modelling...
    The differential equation should be:
    [tex]\frac{a+bsinwt-c\sqrt{H}}{k}=\frac{dH}{dt}[/tex]
    which again I don't know how to solve...
    I simply can't separate it...
    please help..
     
  7. Jan 28, 2006 #6
    I don't know how to do this one but I know the answer using Mathematica

    H[t]= a t w/k - c t w/k H^1/2 + k w/k (Constant of integration)- b Cos[wt]/k

    blumfeld0
     
  8. Jan 28, 2006 #7
    Hi blumfeld0,
    is your answer for the amended ODE I just posted?
     
  9. Jan 28, 2006 #8
    to confirm, is the solution:
    [tex]H(t)=\frac{atw}{k}-ct\frac{w}{k}\sqrt{H}+\frac{kw}{k}(constant of integration)-bcos\frac{wt}{k}[/tex]

    but the "k" can be cancelled?
     
  10. Jan 28, 2006 #9
    can anyone help confirm the answer?
     
  11. Jan 30, 2006 #10

    saltydog

    User Avatar
    Science Advisor
    Homework Helper

    Mathematica 5.2 cannot solve this:

    Code (Text):

    [tex]
    \text{DSolve[}h^{'}[t]==\frac{a+b Sin[\omega t]-c\sqrt{h[t]}}{k},h,t]
    [/tex]
     
    So Blumfeld, how did you arrive at that expression?

    Also, when I back-substitute yours into the ODE, Mathematica does not indicate it satisfies the ODE.
     
    Last edited: Jan 30, 2006
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