Calculate maximal interval by bounding f(t, x)x′=f(t,x)x(t0)=x0

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Calculate maximal interval by bounding f(t, x)x′=f(t,x)x(t0)=x0

clugiarh0j · Accepted Answer

Step 1Consider the solution curvesy(t)=ϕ+(t;t0,y0) of y′=g(t,y).Then the inequality f≤g tells you that at every point the vector field for f crosses the curve of ϕ+ at that point downwards.Thus for any y0&amp;gt;x0, the solution x(t) can not crossϕ+(t;t0,y0) for t≥0.In the limit, alsoy+(t)=ϕ+(t;t0,x0)is an upper bound.One can also put this in formulas by consideringh(t)=ϕ+(t0;t,x(t)),going forward with f and then backward with g, and computing its derivative using the derivation laws of the flow. This should give a falling function. See Dynamical systems proof that f(t) is less than or equal to g(t)The same goes for the lower bound, and also for negative times where the positions of ϕ+ and ϕ− are switched.

Calculate maximal interval by bounding \(\displaystyle{f{{\left({t},\ {x}\right)}}}\)

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