445/545
TEST # 1
Study Guide
Problem #1: Solve an initial value problem for a first order linear system of o.d.e.'s
(a)-[5 pts.]-Rewrite the following system in matrix-vector form: x'  ax by, y'  cx  dy . a,b,c and d
are given constants. Assume that the eigenvalues of the matrix of the given system are real and distinct.
(b)-[10 pts.]-Find a linearly independent pair of vector solutions Y1(t), Y2 (t) of the given system that
 v1 
t
have the form Y (t )  e V , V =   .
v 2 
(c)-[5 pts.]-Use the results of (b) to construct the general solution of the given system.
 x0 
(d)-[5 pts.]-Find the values of the arbitrary constants in the general solution so that Y (0)    .
 y0 
______________________________________________________________________________________
Problem #2: Solve an initial value problem for a first order linear system of o.d.e's
(a)-[3 pts.]-Rewrite the following system in matrix-vector form: x'  ax by, y'  cx  dy .
a,b,c and d are given constants. Assume that the eigenvalues of the matrix of the given system are complex.
(b)-[6 pts.]-Find a linearly independent pair of vector solutions Y1(t), Y2 (t) of the given system that have
 v1 
t
the form Y (t )  e V , V =   .
v 2 
(c)-[5 pts.]-Express the general solution in terms of Y1(t) and Y2 (t) .
(d)-[4 pts.]-Find Re Y1(t) and ImY1(t) .
(e)-[3 pts.]-Express the general solution in terms of Re Y1(t) and ImY1(t) .
(f)-[4 pts.]-Classify the equilibrium point (0, 0) of the given system.
_____________________________________________________________________________________
Problem #3: Solve an initial value problem for a first order linear system of o.d.e's
(a)-[4 pts.]-Re-write the following system in matrix-vector form: x'  ax by, y'  cx  dy .
a,b,c and d are given constants. Assume that the matrix of the given system has exactly one real eigenvalue.
 v1 
t
(b)-[5 pts.]-Find a vector solution Y1(t) of the given system that has the form Y (t )  e V , V =   .
v 2 
(c)-[5 pts.]-Find a vector solution Y2 (t) of the given system that has the form
v 
w 
Y (t )  e t W  tV , V =  1 , W =  1  .
v 2 
w 2 
(d)-[4 pts.]-Express the general solution in terms of Y1(t) and Y2 (t) .
 x0 
(e)-[4 pts.]-Find the values of the arbitrary constants in the general solution so that Y (0)    .
 y0 
(f)-[3 pts.]-Classify the equilibrium point (0, 0) of the given system.
_______________________________________________________________________________________
CONTINUED ON NEXT PAGE
C. O. Bloom
445/545
TEST # 1
Study Guide
Problem #4: Solve an initial value problem for a 3-dimensional first order linear system of o.d.e's
(a)-[5 pts.]-Rewrite the following system in matrix vector form:
x '  ax  by  cz, y '  dx  ey  fz, z'  gx  hy  kz
where a, b, c, d, e, f , g, h, k are given constants. Assume that the eigenvalues of the matrix of the given
system are real and distinct.
(b)-[15 pts.]-Find 3 linearly independent vector solutions Y1 (t ), Y2 (t ), Y3 (t ) of the given system
that have the form
 v1 
t
Y (t )  e V , V = v 2 
 v 3 
(c)-[5 pts.]- Use the result of part (a) to find the solution of the given system that satisfies the initial condition
x0 
Y (0)   y0 
 z 0 
where x0, y0 and z0 are given constants.
_______________________________________________________________________________________
Problem #5:
Consider the following 4 –dimensional linear system x  Ax . Let λ1 and λ2 be the distinct,
real eigenvalues of A with algebraic multiplicities n1 and n2 , and geometric multiplicities
p1 and p2 .
(1) Let u11, u21 ,..., un11 be linearly independent set of eigenvectors, and generalized eigenvector belonging to
the eigenvalue 1 , and let u12 , u22 ,..., un2 2 be linearly independent set of eigenvectors, and generalized
eigenvectors belonging to the eigenvalue λ2 .
At
At
At
λt
At
At
At
(a)-[15 pts]- Express e u11, e u21, ..., e un11 , and e u12 , e u22 ,..., e un2 2 in the form e Pk ( t ) where Pk ( t )
is a polynomial of degree k (0  k  n  1) where n is the algebraic multiplicity of  .
At
At
At
(b)-[5 pts.]-Prove that e Atu11, e Atu21, ..., e Atun11 ,and e u12 , e u22 ,..., e un2 2 are linearly independent.
using the fact that u11, u21 ,..., un11 and u12 , u22 ,..., un2 2 are linearly independent..
(c)-[5 pts.]-Write down the general solution of the given system.
_______________________________________________________________________________________
CONTINUED ON NEXT PAGE
445/545
TEST # 1
Study Guide
Problem #6:
Let P  [u1 , u2 , u3 , u4 ] where
 a1 
 b1 
 c1 
 d1 
a 
b 
c 
d 
u1   2 , u2   2 , u3   2 , u4   2 
 a3 
b3 
c3 
d3 
 
 
 
 
 a4 
b4 
c4 
d 4 
are linearly independent set of generalized eigenvectors corresponding to the eigenvalues of the matrix
 a11 a12 a13 a14 
a
a 22 a 23 a 24 
21

A
.
a 31 a 32 a 33 a 34 


a 41 a 42 a 43 a 44 
1
(a)-[20 pts.]-Given P 1 calculate the columns of the fundamental matrix e At , i.e. calculate PeP APt P 1 .
(b)-[5 pts.]-Use the result of part (a) to find the solution of the initial value problem x  Ax, x( 0 )  x0 . .
____________________________________________________________________________________
Problem #7:
Finding stable and unstable manifolds
(a)-[10 pts.]-Solve the following initial value problem
x '   x, y '  y  x 2 , z '  3z  xy,
x(0)  x0 , y (0)  y0 , z (0)  z0 .
x'  1 x, y '  2 y, z '  3 z  f ( x, y ),
x(0)  x0 , y (0)  y0 , z (0)  z0
where 1 , 2 ,  0, and 3  0 . Also f ( x, y)  O( x 2  y 2 ) as (x,y)  ( 0,0 ).
(b)-[5 pts.]-Use the result of part (a) to find an equation of the stable manifold for the given system.
(c)-[5 pts.]- Use the result of part (a) to find the equation of the unstable manifold for the given system.
(d)-[5 pts.]-Use the results of parts (a) and (b) to prove that the stable manifold is positively invariant.
____________________________________________________________________________________
C. O. Bloom
C. O. Bloom