Find the Solution for x' = t + x/(1+x^2) with Initial Condition x(0) = 0

[twoflower]

Hi all,

I found this problem on web and though we haven't encountered this kind of problem in class I wonder how could it be done. Here it is:


Guess solution of the problem


$$x' = t + \frac{x}{1+x^2},\mbox{ x(0) = 0}$$

ie. find functions $\omega(t) \leq x(t) \leq \phi(t)$ for each $t$ from domain of solution.

Well, I have no clue. I have the results here if anyone tries...

Thank you for hints!

 

[Tide]

HINT: What are the largest and smallest possible values of the real function

$$\frac {x}{1+x^2}$$
?

 

[twoflower]

HINT: What are the largest and smallest possible values of the real function
$$\frac {x}{1+x^2}$$
?

$$\frac{1}{2}$$

and

$$-\frac{1}{2}$$ ?

 

[HallsofIvy]

Yes, and if you replace the "x" in the right hand side of the equation by those, can you then solve for x(t)?

 

[twoflower]

Yes, and if you replace the "x" in the right hand side of the equation by those, can you then solve for x(t)?

Thank you HallsofIvy, so I wrote the bounds

$$x \leq \frac{t^2}{2} + \frac{t}{2}$$

and

$$x \geq \frac{t^2}{2} - \frac{t}{2}$$

It's ok, isn't it? Anyway, the official results say something slightly different...

 

[twoflower]

It's ok, isn't it? Anyway, the official results say something slightly different...

Ok, I have it. In the results, there is

$$\frac{t^2}{2} - t \leq x(t) \leq \frac{t^2}{2} + t$$

which is just result of rougher bounds.

 

[saltydog]

So can someone suggess a method for solving the original ODE analytically? I can't.

 

[HallsofIvy]

Non-linear differential equations, such as this one, tend NOT to have solutions that can be found exactly.

 

[saltydog]

Non-linear differential equations, such as this one, tend NOT to have solutions that can be found exactly.

Hello Hall. How are you? May I say I'm not satisfied by this outcome? I know what you're thinkin': "why do I even bother; he's a pain in the . . .". Tell you what, suppose an asteriod was heading here and we had to find some analytical expression for this ODE in order to successfully deflect it. What progress could the combined intellect of the world muster to do so? I bet a whole dollar something could be done.