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

[saxm]

Hi,

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

f(t) = Ae^{t}+Bte^{t}

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.

 

[lurflurf]

You will need a function such as the product-log to do that.

 

[coomast]

Hi,

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

f(t) = Ae^{t}+Bte^{t}

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:

f=\alpha=(A+Bt)e^t

from which:

g=\alpha-(A+Bt)e^t=0

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

g'=-(A+B+Bt)e^t

The iterative scheme is now:

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

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

0

1

0.701213

0.622262

0.617657

0.617642

Hope this helps,

coomast

 

[element4]

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

see http://mathworld.wolfram.com/LambertW-Function.html" .