Problem: y'=((x-1)/(x^2))*(y^2) , y(1)=1 . Find solutions satisfying the initial condition, and determine the intervals where they exist and where they are unique. Attempt at solution: Let f(x,y)=((x-1)/(x^2))*(y^2), which is continuous near any (x0,y0) provided x0≠0 so a solution with y(x0)=y0 is guaranteed to exist when x0≠0 The partial derivative with respect to y of f(x,y) is 2y*((x-1)/(x^2)) which is continuous near any (x0,y0) provided x0≠0 so a solution with y(x0)=y0 is unique when x0≠0 (By Picard's Theorem) From this the interval in which the solutions are unique is x∈(-∞,0)∪(0,∞). Solving the differential equation and using the initial condition y(1)=1 we see that y(x)=x/(ln|x|+1). The interval of existence is 1/e < x < ∞ Is this right, or...? Thanks for any help!