I'm not sure I would say that you use integration by parts. The differential equation you have there yields to separation of variables thus:
subject to Thus,
and hence
Can you continue from here?
Apologies in advance, I'm rusty and having some trouble and really want to make more of an effort to keep up with my current year.
Have a simple differential:
dx/dt = x^3
Initial condition x(0) = x0
Apparently a simple integration by parts yields:
x(t) = x0 / (1-2x0^2 t)^1/2
I know i'm falling at something trivial and just not recognising the form. Could someone explain this expression? (how has it retained t terms?)
Thank you for bearing with me. Once I get the ball rolling, maybe I can come up with better questions
erm....
t = -1/2x^2 + c
1/x^2 = 2c -2t
x^2 = 1/ (C-2t)
x = 1/ sqt(C-2t)
x(0) = xsubscript0
which gives: C = 1/ (xsub0 ^2)
substitute:
x = 1/ sqt(1/xsub0^2 - 2t)
I think the last step i got confused mostly:
multiply by xsub0^2(rooted) on RHS top and bottom
x = x0 / sqt(1-2x0^2 x t)