Approximate solution of differential equation

1. Nov 7, 2015

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?

2. Nov 8, 2015

HallsofIvy

Staff Emeritus
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.

3. Nov 8, 2015

Staff: Mentor

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.

4. Nov 8, 2015

zhanhai

To HallsofIvy: