ORDINARY DIFFERENTIAL EQUATION
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ODETest3.pdf
Math 2306 Ordinary Diff. Eqns. Name (print): Spring 2020 Test 3
Upload the completed test to D2L by 11:59pm ET, 04/20/2020
Sections: 3.2, 3.3, 3.4, 3.5, 3.6
This test is open book and open notes, and it contains 8 pages (including this cover page) and 4 problems.
You are required to show your work on each problem or reason your answer on this exam.
The following rules apply:
• Organize your work, in a reasonably neat and coherent way, in the space provided. Work scat- tered all over the page without a clear ordering will receive very little credit.
• Mysterious or unsupported answers will not receive full credit. A correct answer, unsup- ported by calculations, explanation, or algebraic work will receive no credit; an incorrect answer supported by substantially correct calculations and explanations might still receive partial credit.
Do not write in the table to the right.
Problem Points Score
1 11
2 10
3 14
4 5
Total: 40
Math 2306 Ordinary Diff. Eqns. Test 3 - Page 2 of 8
1. (11 points) Consider the linear system
dx
dt = 2x + 3y
dy
dt = 2x + y
(a) (4 points) Compute the eigenvalues and eigenvectors of the matrix for the system.
(b) (2 points) For each eigenvalue, determine a corresponding straight-line solution.
Math 2306 Ordinary Diff. Eqns. Test 3 - Page 3 of 8
(c) (2 points) Sketch manually the phase portrait for the linear system by drawing the straight-line solutions, equilibrium point(s) as well as other significantly different solu- tion curves. Do NOT use DE Tools or any other technology for this part or your work will receive no credit.
(d) (1 point) Identify the equilibrium point for the system as a spiral source, a spiral sink or a saddle. Reason your answer.
(e) (2 points) Find the general solution.
Math 2306 Ordinary Diff. Eqns. Test 3 - Page 4 of 8
2. (10 points) Consider the linear system
d −→ Y
dt =
( 2 8 −1 −2
) −→ Y , where
−→ Y (t) =
( x(t) y(t)
) .
(a) (2 points) Compute the eigenvalues of the matrix for the system.
(b) (2 points) For one of the eigenvalues in part (a), compute an eigenvector.
Math 2306 Ordinary Diff. Eqns. Test 3 - Page 5 of 8
(c) (2 points) Sketch manually the phase portrait for the linear system by drawing the
straight-line solutions corresponding to corresponding to −→ Y1(t) and
−→ Y2(t), equilibrium
point(s) as well as other significantly different solution curves. Do NOT use DE Tools or any other technology for this part or your work will receive no credit.
(d) (2 points) Determine two solutions −→ Y1(t) and
−→ Y2(t) for which
−→ Y1(0) and
−→ Y2(0) are linearly
independent.
(e) (2 points) Determine the solution of the linear system subject to the initial condition
−→ Y (0) =
( 2 −1
) .
Math 2306 Ordinary Diff. Eqns. Test 3 - Page 6 of 8
3. (14 points) Consider the initial-value problem
d −→ Y
dt =
( 2 4 −1 6
) −→ Y ,
−→ Y (0) =
( −1
6
) .
(a) (2 points) Compute the eigenvalue(s) of the matrix for the system.
(b) (2 points) Compute an eigenvector corresponding to the eigenvalue found in part (a).
(c) (2 points) Sketch manually the phase portrait, including the solution curve with the
initial condition −→ Y (0) =
( −1
6
) and equilibrium solution (s). Do NOT use DE Tools or
any other technology for this part or your work will receive no credit.
(d) (2 points) Sketch roughly x(t)− and y(t)− graphs of the solution by using the phase portrait in the part (c).
Math 2306 Ordinary Diff. Eqns. Test 3 - Page 7 of 8
(e) (2 points) Find the general solution.
(f) (2 points) Find the particular solution for the initial condition −→ Y (0) =
( −1
6
) .
(g) (2 points) Sketch x(t)− and y(t)− graphs of the solution found in part (g). You may use any technology of your choice and reproduce the graph here.
Math 2306 Ordinary Diff. Eqns. Test 3 - Page 8 of 8
4. (5 points) Find the solution to the initial-value problem
y′′ + 2y′ − 3y = 0, y(0) = 6, y′(0) = −2.
