Local existence and finite-time obstruction for $x'=x^2+t^2$

Consider \[ x'=x^2+t^2,\qquad x(0)=1. \]

1. Prove that there is a solution on \([0,b]\) for some \(b>0\).
2. Give an explicit such \(b\).
3. Find \(c>0\) such that no solution exists on \([0,c]\).