By Gerald Teschl
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Additional info for Ordinary Differential Equations and Dynamical Systems (draft)
R .................... ✲ ... .......... ........ . . ....... . . .. ...... . x(t) ... . .... .... II .... ... Similarly, solutions can only get from III to II but not from II to III. This already has important consequences for the solutions: • For solutions starting in region I there are two cases; either the solution stays in I for all time and hence must converge to +∞ (maybe in finite time) or it enters region II. ). Since it must stay above x = −t this cannot happen in finite time.
38) In the case β = 0 the right-hand side has to be replaced by its limit ψ(t) ≤ α + γt. Of course this last inequality does not provide any new insight. Now we can show that our IVP is well-posed. 8. Suppose f, g ∈ C(U, Rn ) and let f be locally Lipschitz continuous in the second argument, uniformly with respect to the first. 41) (t,x)∈V with V ⊂ U some set containing the graphs of x(t) and y(t). Proof. Without restriction we set t0 = 0. Then we have t |x(t) − y(t)| ≤ |x0 − y0 | + |f (s, x(s)) − g(s, y(s))|ds.
All solutions below x0 (t) converge to the line x = −t. It is clear that similar considerations can be applied to any first-order equation x˙ = f (t, x) and one usually can obtain a quite complete picture of the solutions. However, it is important to point out that the reason for our success was the fact that our equation lives in two dimensions (t, x) ∈ R2 . If we consider higher order equations or systems of equations, we need more dimensions. At first sight this seems only to imply that we can no longer plot everything, but there is another more severe difference: In R2 a curve splits our space into two regions: one above and one below the curve.
Ordinary Differential Equations and Dynamical Systems (draft) by Gerald Teschl