Special case of nonlinear first order ordinary differential equation.

  • Thread starter sola maths
  • Start date
Hi there,

I've having problems solving a particular nonlinear ODE. Any help/suggestions will be highly appreciated.

The nonlinear ODE is:

v[t]*v'[t] + (4*v[t])/(t^2 - 1) = t/(t^2 - 1)

Thank you.
 
1,786
53
If:

[tex]z=\int f(x)dx[/tex]

then:

[tex]\frac{dz}{dx}=f(x)[/tex]

or:

[tex]\frac{dx}{dz}=\frac{1}{f}[/tex]

now just substitute back into the original DE
 
Thanks for your reply, I understand your point but if you look at the example in the link I posted, you will see that the substitution is not as straightforward as you put it.

Seems to me that there's some sort of parametrization to be done.
 
1,786
53
. . . what did I get myself into. But that's ok, need to be willing to try things in math to succeed and not let the risk of failure stop you but I'd rather not make two mistakes in one day. So I think it's this below but I'd have to work on it more with real problems to verify it ok.


Suppose you have:

[tex]vv'=f(t)v+g(t)[/tex]

and you let:

[tex]z=\int f dt[/tex]

to obtain:

[tex]v\frac{dv}{dz}=v+\Phi(t)[/tex]

where:

[tex]\Phi(t)=\frac{g(t)}{f(t)}[/tex]

and suppose we're able to solve:

[tex]v\frac{dv}{dz}=v+\Phi(z)=v+\frac{g(z)}{f(z)}[/tex]

say for example the solution is:

[tex]v(z)=z^3-3\sin(z)[/tex]

but z is paramaterized by t so that we can write the solution for the original problem as:

[tex]v(t)=z^3-3\sin(z);\quad z=\int f(t)[/tex]

I'm not sure at this point but it's what I'm going with for now until I can verify that or someone can correct me.

Also, I should mention this holds for the simple case:

[tex]vv'=tv+t[/tex]

which can be solved exactly but that's still no guarantee my explanation above is correct for the general case. Never worked on this type of problem before
 
Last edited:
1,786
53
Ok, after reviewing I don't think that's correct. If we have:

[tex]y\frac{dy}{dz}=y+\Phi(t)=y+\frac{g(t)}{f(t)}[/tex]

and:

[tex]z=\int f(t)=h(t)[/tex]

then I believe we'd have to compute the inverse:

[tex]t=h^{-1}(z)[/tex]

then solve:

[tex]y\frac{dy}{dz}=y+\Phi\big(h^{-1}(z)\big)[/tex]

I'm probably making a mess out of this but I'd at least like to demonstrate how sometimes real math is not so neat and pretty and quick and often fought with many wrong turns and maybe I'm still going wrong.
 
Pheeww! Mathematics indeed could be rattling, been on the problem for sometime now.
The problem I foresee with your last review is computing:

[tex]t=h^{-1}(z)[/tex]

Can't see how that will be done.
 
1,786
53
Well I probably don't have it exactly right yet. However if this was my problem, I would go to the library and find one of those Russian texts cited in the Eqnworld reference and get some simple test cases with the answers and study them and maybe the references would explain how the parameterization is used. I'll look at it more later.
 

Related Threads for: Special case of nonlinear first order ordinary differential equation.

Replies
1
Views
1K
Replies
6
Views
4K
Replies
11
Views
19K
Replies
8
Views
681
Replies
8
Views
1K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving

Hot Threads

Top