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Results tagged with ca.classical-analysis-and-odes
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user 110094
Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.
6
votes
1
answer
2k
views
Gronwall's inequality for higher order derivatives
Gronwall's inequality says that solutions to the initial value problem $u'(t) \leq \beta(t)u(t)$ with $u(0)=u_0$ are bounded by solutions to the problem with inequality replaced with equality for $t\i …
- 205
4
votes
1
answer
514
views
Controlling subsolutions of a second order linear ODE
Let $f:[0,\infty) \to \mathbb{R}$ obey the differential inequality
$$f'' - 2\alpha f' + 2\alpha f \leq 0$$
where $0 < \alpha < 2$ is some constant. If $f(0) = 0$ and $f'(0) = 1$, can I say that $f(x) …
- 205
0
votes
Controlling solutions of a second order linear differential inequality
Wow, thank you for this detailed solution.
I have a questions about case 1. You say $r'(x)=p_j(r(x)) = \rho_j^{-1}(r(x))$ where $\rho_j=r|_{I_j}$.
In other words, if we restrict our attention to $I_ …
- 205