Download Assignment 3 Solutions - Application Order Differential Equations | MATH 0266A and more Assignments Mathematics in PDF only on Docsity!
Jeffrey Hellrung Friday, October 20, 2006 Math 266A, Homework 02
- Show that for any matrices A, B of small size,
e
(^12) A eB^ e
(^12) A − eA+B^ = O(|A|^3 + |B|^3 ).
Solution By definition,
e
1 2 A^ = I +
A +
A^2 + O(|A|^3 );
eB^ = I + B +
B^2 + O(|B|^3 );
eA+B^ = I + (A + B) +
(A + B)^2 + O(|A + B|^3 ).
It follows that
e
(^12) A eB^ e
(^12) A
I +
A +
A^2
I + B +
B^2
I +
A +
A^2
+ O(|A|^3 + |B|^3 )
= I +
A + B +
A^2 +
AB +
BA +
B^2 + O(|A|^3 + |B|^3 )
= I + A + B +
(A^2 + AB + BA + B^2 ) + O(|A|^3 + |B|^3 )
= eA+B^ + O(|A|^3 + |B|^3 ),
as desired.
- Let
A =
− 2 ǫ 1 0 −ǫ
, x 0 =
Let x(t) solve xt = Ax, x(0) = x 0. Find max 0 ≤t<∞ |x(t)|.
Solution The general solution is easily found to be
x(t) = c 1
e−^2 ǫt^ + c 2
ǫ
e−ǫt.
With the initial condition x(0) = x 0 , we find that
c 1 = −
ǫ , c 2 =
ǫ
giving x(t) =
ǫ
e−ǫt
1 − e−ǫt ǫ
From the above we can easily deduce that
max |x(t)| ≥ |x(1/ǫ)| ∈ O(1/ǫ).
- Let M be a real 2 × 2 matrix with eigenvalues ±i, and consider the ODE
Ut = M U.
Show the orbits in the phase plane are ellipses. Hint: Write M = C−^1 ΛC with Λ = diag(i, −i). Consider V = CU. Find an ODE for V and show that its orbits are circles. Solution As per the hint, we write M = C−^1 ΛC and set V = CU , so that
Vt = CUt = CM U = ΛCU = ΛV.
The solution to this (decoupled) ODE is
V (t) =
c 1 eit c 2 e−it
for some c 1 , c 2 ∈ C. It follows that the real and imaginary parts of each component of U = C−^1 V is a linear combination of cos t and sin t, hence the orbits of Ut = M U are ellipses.
- Consider the ODE (^) ( u v
t
u v
Find and classify its stationary points. Solution Let A =
A is nonsingular (det A = 3 6 = 0), so
)T
is the only stationary point for the ODE. The eigenvalues of A are the roots of
pA(λ) = det(λI − A) = λ^2 + 3,
giving eigenvalues λ± = ±i
- It follows that
)T
is a center.
