Consider an autonomous differential equation i = f(x), where f : Rd → Rd is locally Lipschitz continuous. (i) Does this differential equation have unique local solutions for every initial condition of the form x(0) = xo, where xo E Rª? Justify your answer. ii) Prove that for all yo E Rº, there exist T > 0 and xo E Rd such that there exists a solution A:I→ Rd to this differential equation with A(0) = xo and A(T) = yo-

Consider an autonomous differential equation i = f(x), where f : Rd → Rd is locally Lipschitz continuous. (i) Does this differential equation have unique local solutions for every initial condition of the form x(0) = xo, where xo E Rª? Justify your answer. ii) Prove that for all yo E Rº, there exist T > 0 and xo E Rd such that there exists a solution A:I→ Rd to this differential equation with A(0) = xo and A(T) = yo-

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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