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Math Help - IC's and DE checking Query

  1. #1
    Senior Member bugatti79's Avatar
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    IC's and DE checking Query

    Hi Folks,

    When solving DE's or PDE's is it critical that 2 conditions be met in order to know whether the solution is correct?

    1) Check that the particular solution matches the IC's
    2) Differentiate the particular solution and sub into the original DE

    Just because the particular solution matches the IC's 'alone' doesnt necessarily mean it is the correct solution..? Is this right?

    Thanks
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  2. #2
    A Plied Mathematician
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    Actually, I typically use the homogeneous solution in tandem with the particular solution in order to satisfy the IC's. The particular solution, by definition, has no arbitrary constants, whereas the homogeneous solution does. I have gotten into the habit of always differentiating my general solution (homogeneous plus particular, in the case of linear equations) to check my answer.
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  3. #3
    Senior Member bugatti79's Avatar
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    Quote Originally Posted by Ackbeet View Post
    Actually, I typically use the homogeneous solution in tandem with the particular solution in order to satisfy the IC's. The particular solution, by definition, has no arbitrary constants, whereas the homogeneous solution does. I have gotten into the habit of always differentiating my general solution (homogeneous plus particular, in the case of linear equations) to check my answer.
    Ackbeet,

    I am confused now, how would one differentiate the general solution since in the case of PDE's it would be of the form u(x,y)=f(x,y) etc? Wouldnt you need the particular.

    Also, does homogenous DE related to ordinary differential eqns only?

    Thanks
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  4. #4
    A Plied Mathematician
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    The general solution, for linear non-homogeneous DE's, whether ODE or PDE (PDE's can be homogeneous or non-homogeneous, just like ODE's), consists of the homogeneous solution added to a particular solution. You just use the standard derivative of a sum rule from calculus to differentiate the general solution.

    Here's a linear non-homogeneous PDE:

    u_{t}=ku_{xx}+g(t),

    the heat equation with a time-dependent forcing function.
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  5. #5
    Senior Member bugatti79's Avatar
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    Quote Originally Posted by Ackbeet View Post
    The general solution, for linear non-homogeneous DE's, whether ODE or PDE (PDE's can be homogeneous or non-homogeneous, just like ODE's), consists of the homogeneous solution added to a particular solution. You just use the standard derivative of a sum rule from calculus to differentiate the general solution.

    Here's a linear non-homogeneous PDE:

    u_{t}=ku_{xx}+g(t),

    the heat equation with a time-dependent forcing function.
    Ok, what you have given is also a second order PDE.
    Standard deriviate of a sum rule? What is this?
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  6. #6
    A Plied Mathematician
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    From calculus, you know that if f(x) and g(x) are both differentiable, then

    \dfrac{d}{dx}(f(x)+g(x))=\dfrac{df(x)}{dx}+\dfrac{  dg(x)}{dx}, right? In this context, if I'm checking that a candidate general solution actually solves a DE, I need to differentiate the sum y=y_{c}+y_{p} as

    y'=y_{c}'+y_{p}'.

    That's all I mean. But it's y, not y_{c} or y_{p} that I really need to plug into the DE to see if it solves the DE.
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  7. #7
    Senior Member bugatti79's Avatar
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    Thanks alot, that makes sense now!
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  8. #8
    A Plied Mathematician
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    You're very welcome!
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