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# We begin some more generalities related to first order ODE

Last updated: 9/18/2023 We begin some more generalities related to first order ODE As in the preceding part I denoted an open interval and an open subset of R We consider a function f Ixn R giving rise to the first order ODE dX dt f t x dX dt Note that for each tEI held fixed the map ft R x f x is a vector field on 2 Interpreting as the time variable we call the function 7 a time dependant vector field on And often we call a solution X J of the ODE 8 an integral curve of the vector field f An initial condition for the ODE 8 consists of a pair oo with to El x En and the associated initial value problem is f t X x t Xo X J N Finally recall that a solution of 9 is an at least once continuously differentiable curve J being an open interval with t EJ CI satisfying dX 1 dt MATHEMATICS Proposition 1 In the context of the IVP 9 we consider yet another equation the following integral equation in an unknown function X J X 1 x ff s x s ds teJ 10 7 f t X t for all tEI and the initial condition X to x Following result relates solutions of the IVP 9 and those of the integral equation 10 X 1 x f s X s ds 8 x 0 X A continuously differentiable curve X J J being an open subinterval of I with t EJ is a solution of the IVP 9 if and only if it satisfies the integral equation 10 Proof I First suppose that the curve X J satisfies the integral equation 10 Putting t t in 10 we get 10 0 9 7 d Thus X satisfies the initial condition Next differentiating 10 we get dX t dt ff s X s ds f 1 X 1 by fundamental theorem of integral calculus Above me hare verified that a solution th the integral