Finding Solutions for Non-Linear Equations and ODEs in Math

[eljose]

Let,s suppose I'm asked to find a function with certain properties in math, let's call this function f(x), my question is if i find that f(x) must satisfy a certain differential or integral equation let's say:

$$a(x)f''+b(x)(f')^2 + c(x)tan(f) = 0$$ (NOn- linear ODE )

$$x+ f(x)= \int_a ^ b dy log(y^2 +f(x) )$$ (Non-linear equation)

The question is...does this mean that the function f(x) as a solution of an ODE or a Non-linear integral equation necessarily exist?...

 

[HallsofIvy]

I'm not sure I understand your question. If you mean you have actually have found a function satisfying a given equation, then, yes, it exists!

If you mean you have been asked to find a function satisfying a given equation, then, no, it is not necessarily true that such a function exists. (There are a variety of "existence and uniqueness theorems" that assert that such-and-such problems have solutions. I don't know of any that assert that every non-linear differential equation or every non-linear integral equation has a solution, A perfectly good answer to such a question is "no such function exists" and then, of course, proving there is no such function. Of course, actually finding the function is itself proof that such a function does exist!

 

[eljose]

but the question (from my point of view) would be:

-How could you know that a function satisfying:

$$x+ f(x) = \int_a ^b dy Log (y^2 +f(x))$$ exist?...well the question is that you can always use a "Numerical method" ( integration by quadratures, and all that) so you can "draw" a picture of how the function would look like , and you can check that the function exists and it's Non-zero.

 

[HallsofIvy]

Except that a numerical solution is at best "approximate". And remember that the approximation is saying that the function given by the numerical solution approximately satisfies the equation. It is quite possible that, even if an equation has an approximate numerical solution, there is no exact solution.

 

[WigneRacah]

but the question (from my point of view) would be:

-How could you know that a function satisfying:

$$x+ f(x) = \int_a ^b dy Log (y^2 +f(x))$$ exist?...well the question is that you can always use a "Numerical method" ( integration by quadratures, and all that) so you can "draw" a picture of how the function would look like , and you can check that the function exists and it's Non-zero.

It may be useful the following:

$$\int dy Log(y^2 + f(x)) = y Log(y^2 + f(x)) - 2 \int dy \frac{y^2}{y^2 + f(x)}$$

(by parts).