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Approximate solution of differential equation

  1. Nov 7, 2015 #1
    Differential equation: F(y'',y',y,x)=0,

    Now, there is g=g(x) with F(g'',g',g,x)=δ, where δ is small. Then, can g(x) be taken as an approximate solution of F(y'',y',y,x)=0?
  2. jcsd
  3. Nov 8, 2015 #2


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    That depends strongly on F. As long as F is "well behaved", g making F close to 0 will itself be close to y that makes F equal to 0. However, there will be some functions, that are not continuous or not differentiable or not "sufficiently differentiable", such that this is not true. That is, that a function, g, that makes F small may be wildly different from y that makes F 0.
  4. Nov 8, 2015 #3


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    y''+y=0 has the solution y=sin(x). It also has the solution y=100*sin(x) which is completely different.
    Even without δ, you can get wildly different results. You have to fix initial conditions to get something like that.
  5. Nov 8, 2015 #4
    To HallsofIvy:

    Thank you for your reply!

    Suppose F is similar to Sturm-Liouville form:
    where p(x) has no more than first-order singularity point, and q(x) and h(x) each actually has no singularity; then, is F "well behaved"? And, in such case, would the approximation be valid for the only singularity point (x=0)?
    Last edited: Nov 8, 2015
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