Hello to all of you
I hope, that in English it will easier for me to write my question.
And here it is - I have solve an ordinary differential equation, by use of "finite difference method". This equation describes the velocity of a freely fallin spherical drop as a fonction of a "contact time".
The symbol for velocity is "U".
Contact time is "t".
Here is the equation:
![$\frac{dU}{dt} = 9.7819 - 0.3762U^2$ $\frac{dU}{dt} = 9.7819 - 0.3762U^2$](https://dxdy-03.korotkov.co.uk/f/2/f/2/2f264afdd4b5807a6701449c5191a1b682.png)
I'use the forward difference approximation:
![$\frac{U_{i+1} - U_i} {h} = 9.7819 - 0.3762U^2_i$ $\frac{U_{i+1} - U_i} {h} = 9.7819 - 0.3762U^2_i$](https://dxdy-03.korotkov.co.uk/f/a/4/e/a4e031ff9889cf9ae6afc538308d425a82.png)
and when I develop it, I obtain:
![$U_{i+1} = 9.7819h - 0.3762hU^2_i + U_i$ $U_{i+1} = 9.7819h - 0.3762hU^2_i + U_i$](https://dxdy-01.korotkov.co.uk/f/4/e/3/4e314ca43c6ec7cfa36c436c8066444b82.png)
then
![$U_{i+1} = 9.7819h + U_i(1 - 0.3762hU_i)$ $U_{i+1} = 9.7819h + U_i(1 - 0.3762hU_i)$](https://dxdy-03.korotkov.co.uk/f/e/a/7/ea7e22a94afb13b580137380d2601ad282.png)
My question is relied to the second degree of U and it's development in finite difference approximation. This equation is going to be resolved by a Fortran program. I've alreday done an algorythm for equations describing radial diffusion in a sphere, so I think, that this one will not be that hard. But I don't get, how to describe this ordinary diff. equation.
Once I do it, I have to implement the "value" of U, into an integral, which gives me a distance. But first of all the equation :)
I know, that this is easy for you, but for me is difficult.
Thank you all!!!