Note: this has been edited to fix a typo. 1. The problem statement, all variables and given/known data The problem is to find the maximum solution on all of R of the differential equation dy/dt = t * y^(1/3) subject to the initial-value conditoin y(1) = -1. 2. Relevant equations 3. The attempt at a solution This equation is not subject to the conclusion of Picard's standard existence and uniqueness theorem since t * y^(1/3) is not Lipschitz in the y variable. But the existence of a solution is guaranteed by Peano's theorem. Further, it's proved in the theory of ODEs that there will be a maximum solution--a solution that is greater than or equal to all others at all points. By separation of variables it's easy to find a particular solution and then use the initial condition to find the value of the constant that emerges in the process. But as for finding the maximum solution I don't know where to begin. Any help would be greatly appreciated!