Flashcards
ODEs Flash Cards
Flashcards do conjunto (8)
- 1What is the Lipschitz Condition?A function f(x,t) satisfies the Lipschitz condition with respect to x in a rectangle D in the (t,x) plane if there is a constant L (the Lipschitz constant) such that |f(x(1),t) - f(x(2),t)|<(or equal to) |x(1)-x(2)|
- 2What is Picard's Existence Theorem?Suppose f is a continuous function that satisfies the Lipschitz condition in some rectangle D containing (x(0),t(0)). Then the IVP, dx/dt=f(x,t), x(t(0)) = x(0), has a unique solution x(t) for t(0)0 (d can be infinity). Doesn't necessarily exist for all t, as the solution may blow up.
- 3Define StabilityWe say x* is stable if a solution x(t) of x'=f(x) that start really close to x*, stays really close to x*. x* is asymptotically stable if ( in addition) x(t)-->x* for solutions x(t) that starts close to x. x* is unstable if it is not stable.
- 4What is the Superposition Principle?(for homogeneous systems only) If (vectors) x(1),...,x(k) are solutions of x'=A(t)x then so is
c(1)x(1) + c(2)x(2) + ... + c(k)x(k) for an constants c(1),...,c(k) - 5Define Linear DependenceVector valued functions x(1),...,x(k) are linearly dependent iff there exists constants c(1),...,c(k) not all zero, s.t.
c(1)x(1) + ... + c(k)x(k) = 0 for all t - 6Define Linear IndependenceVector valued functions x(1),...,x(k) are linearly independent iff
c(1)x(1) + ... + c(k)x(k) = 0 implies that
c(1)=c(2)=...=c(k)=0 - 7What is the Wronskian (for vector valued functions)?Let x(i)(t):r-->R^n be vector valued functions fo t for i-1,...,n. The Wronskian, w(t), of x(1),...,x(n) is defined by:
w(t) = det (x(1)(t) | ... | x(n)(t) ) - 8#what is Abel's Formula for the Wronskian?Let x(1)(t),...,x(n)(t) be solutions of x'=A(t)x. If w(t) is the Wronskian of x(1),...,x(n) then
w(t) = w(t(0)) exp[INTEGRAL(t(o) to t) tr(A(t))dt]
Informações
- Descrição
- Mathematics FlashCards sobre ODEs Flash Cards, criado por Emma B em 15-01-2019.
- Criado por
- Emma B
- Visualizações
- 0
- Avaliação