Ok so ill give an example, x'(t) = log(3t(x(t)2)) is differential equation where t_{0} = 3 and x_{0} = 5
The initial value problem is x(t_{0}) = x_{0}.
So what i'de do is plug into initial value problem to get x(3) = 5, so on a graph this plot would be at (5,3)? Then plop conditions into differential equation so: x'(3) = log(3*3(5  2)) = 1.43 which would be at a plot (1.43,3)? So x'(3) < x(3), does this mean that the solution is unique for all t? If so, why is this? Just want to understand 100%.
