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Violating intial conditions: ODEs

  1. Sep 22, 2015 #1
    Hi Everyone,

    I had a quick question. If you have an IVP ODE and you solve for the general solution first and you had fractions in it, could you multiply by a number to make it "easier" (whole number, rather than involving fractions) without violating the initial conditions?

    Thanks
     
  2. jcsd
  3. Sep 22, 2015 #2

    Geofleur

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    It depends on whether the initial conditions are homogeneous or not. For example, if you had something like ## x(0) = 0 ## and ## \frac{dx}{dt}(0) = 0 ##, you could multiply the solution to the ODE by any nonzero number you want, and it would still satisfy the ICs. But if you had, say ## x(0) = 1 ##, you would have a problem.
     
  4. Sep 23, 2015 #3
    In general, you need to transform your initial conditions as well, for example,

    [itex]y'=x[/itex], [itex]y(0)=1[/itex]

    has as general solution [itex]y=\frac{1}{2}x^2 + C[/itex]
    and C=1 when you use the initial condition.
    You can multiply the general solution by 2 and use the transformation z=2y to get rid of the fractions:
    [itex]z=x^2 + 2C[/itex]
    but you also have to multiply the original ODE by 2 (because dz/dx=2dy/dx) and the initial condition to rewrite it to z:
    [itex]z'=2x[/itex], [itex]z(0)=2[/itex]
     
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