Elementary Autonomous Initial Value Textbook Problem

Solve the following initial value problem and determine the interval where the solution exists.

dy/dt = 2ty^2

y(0) = y_0

Textbook answer is:

When y_0 <> 0, then y = 1/(1/y_0 - t^2).

When y_0 = 0, then y = 0

When y_0 > 0, the interval where solution exists is |t| < 1/sqrt(y_0)

When y_0 <= 0, the interval where solution exists is -infinity < t < +infinity

My work:

The equation is separable:

dy/y^2 = 2t dt

-1/y = t^2 + C

When y_0 = 0, I use intuition to come to the same solution that the textbook shows.

When y_0 <> 0

C = -1/y_0

y = 1/(1/y_0 - t^2).

The interval where the solution exists is where the denominator expression does not equal zero. Algebraically, that simplifies to t <> 1/sqrt(y_0)

The textbook answer is different. What did I do wrong or omit?