Hello and thanks in advance for anyone who can help at all. I have two problems that have stumped me.. I'm in an advanced ODE class. Here they are:(adsbygoogle = window.adsbygoogle || []).push({});

1) Consider the first order ODE f_a(x) where a is a parameter; let f_a(x0) = 0

for some solution x0 and also let f'_a(x0) != 0. Prove that the ODE

f_a+e(x) has an equlibrium point x0(e) where e -> x0(e) is a smooth function satisfying x0(0) = x0 for e sufficiently small.

2) Consider the system X' = F(X) where X is in R_n. Suppose F has an equilbrium point at X0. Show that there exists a change of coordinates that moves X0 to the origin and converts the system to X' = AX + G(X) where A is an nxn matrix which is the canonical form of DF_X0 and where G(X) satifies

lim (|G(X)| / |X|) = 0.

|X|->0

I am so lost on these...can anyone help pleeeeeeeeeease?

Michelle

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# Two ODE proofs

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