Global solution to inhomogeneous Bernoulli ODE

[doobly]

Hi everyone,

I have an inhomogeneous Bernoulli type ODE given by

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

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

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

 

[HallsofIvy]

As long as l(t) and v(t) are "Lipschitz" ("differentiable" is sufficient but not necessary) on [0, 1], that follows from the elementary "existence and uniqueness" theorem for intial value prolems of the for equations of the form y'= f(t, y).