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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 (x_{0},y_{0}) provided x_{0}≠0 so a solution with y(x_{0})=y_{0}is guaranteed to exist when x_{0}≠0

The partial derivative with respect to y of f(x,y) is 2y*((x-1)/(x^2)) which is continuous near any (x_{0},y_{0}) provided x_{0}≠0 so a solution with y(x_{0})=y_{0}is unique when x_{0}≠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!

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# I Interval of existence and uniqueness of a separable 1st ODE

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