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given dy/dx=sqrt(x^2+y^2-9)

and y(a)=b

i)for what values of a and b would a local solution exist and why

ii)for what values of a and b would a unique solution exist and why

for i) i got all values such that a^2+b^2<=9

for ii) i got all values such that a^2+b^2>9

i'm not 100% sure this is correct and would be happy if someone could confirm/show otherwise these results.

secondly

how would one show that the only solution of

y''+ty'+(1+t^2)y^2=0

that touches the t axis at some point (t0,0) is the identically zero solution

thanks for your help