How Do I Solve This Differential Equation Correctly?

[schattenjaeger]

I actually haven't had differential equations, but I managed to somehow to register for the class I am in-_-, fortunately it's easy, but I'm still not sure how

dx/dt = (.8*sqrt(100+x^2)/x

I figure multiply out the dt, so dx=(.8*sqrt(100+x^2)/x dt, then integrate

so x=allthatstuff *t, except I'm not sure that's right, should everything be constant like that?

 

[whozum]

\frac{dx}{dt} = \frac{4}{5}\frac{\sqrt{100+x^2}}{5x}

I THINK you subtract the RHS so then its off the form:

Mx' + Nx = 0

I didnt take diff eq either and don't really know much about it, but if you can take it from there, go for it. My next thought would be some kind of integrating factor.

 

[dextercioby]

What do you know about separation of variables...?

Daniel.

 

[schattenjaeger]

http://pacific.uta.edu/~qiming/Project2.htm


There's the actual problem, I already did the computational part and all that, now I just need to compare my result with the analytic one, which I don't know how to do:(

 

[dextercioby]

It's a simple integration.

Can u compute

\int \frac{\sqrt{x^{2}+100}}{x} \ dx...?


U could use a hyperbolic substitution

x=10\sinh t

Daniel.

 

[schattenjaeger]

err, how did you get that? I wound up with x on top(in which case you can use the sub u=100+x^2

but I haven't had the class yet so I've really got no clue



EDIT: My chief concern is no matter what way I try to solve it, I'm not getting a result that really makes sense with the given problem, and certainly doesn't match my numerical answer.

 

[James R]

You have:

\frac{dx}{dt} = \frac{4}{5}\frac{\sqrt{100+x^2}}{5x}

Therefore:

\int \frac{4}{25}\,dt = \int \frac{x}{\sqrt{100+x^2}} \, dx + c

To do the integral on the right hand side, put u = 100 + x^2. Then

du = 2x dx

and the integral becomes

\int \frac{1}{2\sqrt{u}} \, du

CAn you do it from there?

 

[schattenjaeger]

Yah, that's not the problem anymore

it's 5/4 sqrt(100+x^2) = t, the problem is look at the actual question in that link I posted. The answer doesn't seem to make sense. At least I don't think it does. Well, it doesn't match my numerical answer at least, which means I probably screwed up. *runs off to fix*

well yah, it doesn't make any sense considering the actual question

 

[dextercioby]

Yes,alright,it's that integral written by James R.Mine would have been a little tricky...

Daniel.