Ordinary Differential Equations and Dynamical Systems Vol2 · Problem 3.11

Nonautonomous Linear Systems

Mathematics — Calculus · ivp-existence-interval

Problem

Consider the initial value problem x′=(cos⁡t) x2,x(t0)=x0.x' = (\cos t)\, x^2, \quad x(t_0) = x_0. (a) Find the solution x(t,t0,x0)x(t, t_0, x_0). (b) Prove that if ∣1/x0+sin⁡t0∣>1|1/x_0 + \sin t_0| > 1, then the maximal existence interval is (−∞,∞)(-\infty, \infty). (c) Prove that for every t,s∈Rt, s \in \mathbb{R}, {y∈R:∣1/y+sin⁡s∣>1}⊂U(t,s).\{y \in \mathbb{R} : |1/y + \sin s| > 1\} \subset U(t, s).

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