For the following initial value problem:

Question

dy/dt=1/((y+1)(t-2))
a)Find a formula for the solution.
b) State the domain of definition of the solution.
c) Describe what happens to the solution as it approaches the limit of its domain of definition. Why can't the solution be extended for more time?

I separated and integrated and got y(t)=sqrt(2ln|t-2|+C)-1 and I don't really know where to go from there.

Answer 1

y = √(2ln|t-2|+C)-1

we know that √x is defined only for x >= 0, so we must have

2ln|t-2| + C >= 0
ln|t-2| >= -C/2
t-2 >= e^(-C/2)
t >= 2+e^(-C/2)

In general, t>=2, but the form of the solution suggested that already.

As t -> 2, ln|t-2| -> infinity

Not sure why large t cannot be used. May be missing some of the characteristics of the problem.
The initial value conditons will determine C.