Banach Fixed Point and Differential Equations

[ChemEng1]

Homework Statement

Find the value of x, correct to three decimal places for which: $\int^{x}_{0}\frac{t^{2}}{1+t^{2}}dt=\frac{1}{2}$.

Homework Equations

Banach's Fixed Point Theorem
Picard's Theorem?

The Attempt at a Solution

I'm not sure where to start with this type of problem.

From other BFPT problems, I will need to show a contraction mapping into itself.

Any pointers would be greatly appreciated.

 

[HallsofIvy]

Banach's fixed point theorem and Picard's method show that this problem has a unique solution. Actually finding the solution doesn't require that.

You could, for example, actually integrate that:
[tex]\int_0^x \frac{t^2}{1+ t^2}dt= \int_0^x 1- \frac{1}{t^2+ 1}dt[/itex]
is pretty easy to integrate.

 

[ChemEng1]

$\int^{x}_{0}\frac{t^{2}}{1+t^{2}}dt=\int^{x}_{0}(1-\frac{1}{1+t^{2}})dt$

Let t=tanθ. Then dt=sec²θdθ.

$\int^{x}_{0}(1-\frac{1}{1+t^{2}})dt=\int^{x}_{0}dt-\int^{x}_{0}\frac{1}{sec^{2}\vartheta}$sec²θdθ=$\int^{x}_{0}dt-\int^{x}_{0}$dθ=t-arctan(t) from 0 to x=x-arctan(x).

x-arctan(x)=.5 at x=1.4743.

Are you familiar with any approaches that use BFPT?