Explicitly find the maximal interval I about 0 on which we can solve the following differential equation:
I'm not sure how to approach. I know that if it was
then I can divide both sides and integrate and get
, but there's a square term.
Assume there is a non-vanishing solution on an interval I about 0.
Thus, y' = (y)^2 this means y'/(y^2) = 1, integrating means (1/y) = x+k.
Since y(0)=1 it means 1/y(0) = k and so k =1.
Thus, the solution must be 1/y=x+1.
Thus, if -1 is not in I then we can take reciprocals to get y = 1/(x+1).
Therefore, I has the condition that -1 is not in it.
Hence, the largest interval is (-1, ).