Approximate solution of differential equation

[zhanhai]

Differential equation: F(y'',y',y,x)=0,
y=y(x).

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?

 

[HallsofIvy]

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.

 

[mfb]

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.

 

[zhanhai]

To https://www.physicsforums.com/members/hallsofivy.331/:

Thank you for your reply!

Suppose F is similar to Sturm-Liouville form:
y''+p(x)y'+q(x)y=h(x),
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)?