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doobly

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I have an inhomogeneous Bernoulli type ODE given by

[itex] u'(t) = \kappa u(t) + \ell(t) u^{\gamma}(t) + v(t),\ \ \ u(T)=b>0,...(1) [/itex]

where [itex] t\in[0,T],\ \ \gamma\in (0,1) [/itex].

My concern is that how to prove the existence and uniqueness of the solution u(t) for all [itex]t\in [0,T] .[/itex] Thanks very much.