Proving Bounds for f(x) using Differential Equations

In summary, the problem is to show that for the differential equation f'(x) = 1/(x^2 + f(x)^2) and initial condition f(1) = 1, the solution satisfies |f(x)| <= 5pi/4 when x >= 1. The solution involves integrating the equation and using the derivative of arctan, but it is important to note that the problem does not ask for the explicit solution.
  • #1
singedang2
26
0
differential equation! urgent!

Homework Statement


we have [tex]f'(x) = \frac{1}{x^2 + f(x)^2}[/tex]
and i need to show that [tex]|f(x)| \leq \frac{5 \pi}{4}[/tex] when [tex]x \geq 1[/tex]


Homework Equations





The Attempt at a Solution



i know this has something to do with arctan, (by the looks of f'(x), it looks similar to arctan derivative) but i really don't know how to attempt this. help pls thanks!
 
Last edited:
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  • #2
I don't know how to get a trigonometric funtion in this but if y=f(x), then [tex]dy/dx=f'(x)[/tex]
This implies that [tex]y^2dy=dx/(x^2)[/tex]
Integrating this, you get [tex]y^3/3= -1/x +c[/tex] where c is the constant of integration. Is there some other information given? By derivative of arctan, I guess you mean [tex]d(tan^-1(x))/dx =1/(1+x^2) [/tex]
 
  • #3
sorry, there was mistake in what was written in latex. it's fixed now.
 
  • #4
I presume chaoseverlasting's response was to some incorrectly written equation which has been edited since the equation, as now given, is not separable.

I notice that the problem does not ask you to solve the differential equation, just to show that its solution must satisfy some property. But I wonder if there shouldn't be some initial value given? If the problem were to solve
[tex]f'(x)= \frac{1}{x^2+ f^2(x)}[/tex]
with initial condition f(1)= 4, then since the right hand side is defined and differentiable at (1,4), the problem has a solution in some neighborhood of (1, 4) but clearly does not satisfy "[tex]|f(x)| \leq \frac{5 \pi}{4}< 4[/tex]"
 
Last edited by a moderator:
  • #5
oh right... f(1) = 1 i think this is the given value.
 
  • #6
then does it mean now [tex]|f(x)| \leq \frac{5 \pi}{4} [/tex] ?
how can i show this using that differential equation?
 
  • #7
any help?
 

Related to Proving Bounds for f(x) using Differential Equations

What is a differential equation?

A differential equation is an equation that relates a function with one or more of its derivatives. It is used to model and describe various physical phenomena in science and engineering.

What are the types of differential equations?

There are several types of differential equations, including ordinary differential equations (ODEs), partial differential equations (PDEs), and stochastic differential equations (SDEs). ODEs involve only one independent variable, while PDEs involve multiple independent variables. SDEs incorporate random variables or noise into the equation.

What is the order of a differential equation?

The order of a differential equation is the highest derivative present in the equation. For example, a first-order differential equation contains only first derivatives, while a second-order differential equation contains second derivatives.

How are differential equations solved?

Differential equations can be solved analytically or numerically. Analytical solutions involve finding an exact formula for the solution, while numerical solutions use algorithms to approximate the solution. The method used to solve a differential equation depends on its type and complexity.

What are some applications of differential equations?

Differential equations are used in various fields such as physics, engineering, and economics to model and predict real-world phenomena. They are also used in the development of technologies such as computer graphics, weather forecasting, and medical imaging.

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