How to solve 2nd order ODE solution eg. te^t+e^t, for t?


by saxm
Tags: ode
saxm
saxm is offline
#1
Mar4-09, 07:06 PM
P: 4
Hi,

I have a second order differential equation with a solution in the form:

[tex]f(t) = Ae^{t}+Bte^{t}[/tex]

I want to solve for t, ie. work out for what value of t does the function f(t) have a particular value. But there seems to be no way (that I know of) to do this. Can anyone give me any pointers to what to do here?

Thanks.
Phys.Org News Partner Science news on Phys.org
Cougars' diverse diet helped them survive the Pleistocene mass extinction
Cyber risks can cause disruption on scale of 2008 crisis, study says
Mantis shrimp stronger than airplanes
lurflurf
lurflurf is offline
#2
Mar4-09, 07:52 PM
HW Helper
P: 2,166
You will need a function such as the product-log to do that.
coomast
coomast is offline
#3
Mar5-09, 03:00 PM
P: 279
Quote Quote by saxm View Post
Hi,

I have a second order differential equation with a solution in the form:

[tex]f(t) = Ae^{t}+Bte^{t}[/tex]

I want to solve for t, ie. work out for what value of t does the function f(t) have a particular value. But there seems to be no way (that I know of) to do this. Can anyone give me any pointers to what to do here?

Thanks.
Another way would be to use the Newton-Raphson iterative scheme. Here is a link:

http://nl.wikipedia.org/wiki/Newton-Raphson

Using this for your equation you get:

[tex]f=\alpha=(A+Bt)e^t[/tex]

from which:

[tex]g=\alpha-(A+Bt)e^t=0[/tex]

The function to be solved. The derivative is found to be:

[tex]g'=-(A+B+Bt)e^t[/tex]

The iterative scheme is now:

[tex]t_{n+1}=t_n+\frac{\alpha-(A+Bt)e^t}{(A+B+Bt)e^t}[/tex]

Start with [itex]t_0=0[/itex], giving for the example [itex]A=B=1[/itex], [itex]\alpha=3[/itex]:

[tex]0[/tex]

[tex]1[/tex]

[tex]0.701213[/tex]

[tex]0.622262[/tex]

[tex]0.617657[/tex]

[tex]0.617642[/tex]

Hope this helps,

coomast

element4
element4 is offline
#4
Mar5-09, 04:08 PM
P: 107

How to solve 2nd order ODE solution eg. te^t+e^t, for t?


Maple 12 suggests [tex]t = \text{LambertW}\left( \frac{f\cdot \exp{\frac AB}}B\right) - \frac AB[/tex]

see here.


Register to reply

Related Discussions
Solve the First-Order ODE Differential Equations 1
How to solve 2nd order d.e ? Is this the right start? Calculus & Beyond Homework 6
How do you solve this nonlinear first order DE Differential Equations 13
2nd Order De Solution Differential Equations 2
2nd order DE, is there a way to solve this without series? Differential Equations 11