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