306D
Test #3
Study Guide
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Problem #1: Solving 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)-[5 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 
(e)-[5 pts.]-Use information from part (b) to classify the equilibrium point (0, 0) of the given system.
(See sections 9.1, 9.2)
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Problem #2: Solving an initial value problem for a first order linear system of o.d.e's
(a)-[4 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)-[7 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 
(d)-[5 pts.]-Find Re Y1(t) and ImY1(t) .
(e)-[5 pts.]-Express the general solution of the given system in terms of Re Y1(t) and ImY1(t) .
(e)-[4 pts.]-Classify the equilibrium point (0, 0) of the given system.
(See sections 9.1, 9.2)
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Problem #3: Solving an initial value problem for a first order linear system of o.d.e's
(a)-[3 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)-[4 pts.]-Classify the equilibrium point (0, 0) of the given system.
(See sections 9.1, 9.2)
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C. O. Bloom
306D
Test #3
Study Guide
Problem #4: Solving 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)-[12 pts.]-Find 3 linearly independent pair of 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)-[3 pts.]-Use the results of (b) to construct the general solution of the given system.
(d)-[5 pts.]-Find the values of the arbitrary constants in the general solution so that
x0 
Y (0)   y0 
 z 0 
where x0, y0 and z0 are given constants.
(See section 9.4)
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Problem #5:
Qualitative Behavior of Planar Systems
Consider the planar system x '  Ax where the elements of the matrix A are constants.
(a)-[5 pts.]-Assuming that the eigenvalues 1 and 2 of A are of opposite sign sketch the 4-half line
solutions of the form  e1t v1 ,  e2t v 2 where v1 and v 2 the eigenvectors of A corresponding to 1 and 2 .
(b)-[5 pts.]-Sketch a non-linear solution of x '  Ax in each region between the half line
solutions drawn in part (a).
(c)-[3 pts.]-Classify the equilibrium point (0, 0).
(d)-[5 pts.]- Assuming that the eigenvalues 1 and 2 of A are both negative sketch the 4-half line
solutions of the form  e1t v1 ,  e2t v 2 where v1 and v 2 the eigenvectors of A corresponding to 1 and 2 .
(d)-[5 pts.]- Sketch a non-linear solution of x '  Ax in each region between the half line
solutions drawn in part (a).
(e)-[2 pts.]- Classify the equilibrium point (0, 0).
(See section 9.3)
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