# Guessing the solution

1. Jan 1, 2006

### 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!

2. Jan 1, 2006

### Tide

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

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

3. Jan 1, 2006

### twoflower

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

and

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

4. Jan 1, 2006

### HallsofIvy

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

5. Jan 2, 2006

### twoflower

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...

6. Jan 2, 2006

### twoflower

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.

7. Jan 2, 2006

### saltydog

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

8. Jan 2, 2006

### HallsofIvy

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

9. Jan 2, 2006

### saltydog

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.