Problems listed here are unequivocally worth doing for practice. Those omitted from this list are omitted for a number of possible reasons - some are simple warm-ups that are unlikely to appear on a midterm or exam, some include at least one part that requires a computer or graphing calculator, some are of a slightly more theoretical nature than I don't expect you to be able to do on a midterm of exam and others may have just escaped my search by accident.
Chapter 1
| Section
|
Problems
|
Description
|
| 1.1-1.2
|
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Most of these problems are introductory and would be good if you are confused about some of the basics.
|
| 1.3
|
1-6
|
Classify by order, linearity
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|
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7-14
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Verify that a given function is a solution to particular ODE.
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|
|
19-20
|
Find the value of $r$ so that $t^r$ solves the given ODE.
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Chapter 2
| Section
|
Problems
|
Description
|
| 2.1
|
13-20
|
Problems which can be solved using an integrating factor. Try using Method of Undetermined Coefficients as well for practice.
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|
|
30
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Choosing ICs to ensure boundedness of solutions
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|
|
34-37
|
Construct equations whose solutions have the prescribed asymptotic behaviour.
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| 2.2
|
1-8
|
Separable equations
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| 2.3
|
1-4
|
Salt water inflow/outflow modeling problems
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|
|
17
|
Stephan-Boltzmann heat loss.
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|
|
18
|
First order forcing (temperature).
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Chapter 3
| Section
|
Problems
|
Description
|
| 3.1
|
1-8
|
|
|
|
9-16
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17-18
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|
|
21-24
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|
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|
28
|
|
| 3.2
|
1-6
|
|
|
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24-27
|
|
|
|
28
|
|
| 3.3
|
7-22
|
|
|
|
27
|
|
| 3.4
|
1-14
|
|
|
|
23-30
|
Reduction of order
|
| 3.5
|
1-20
|
|
|
|
30
|
|
| 3.7
|
1-4, 16
|
Converting from sum of sin and cos to a single cos expression.
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|
|
6, 7, 11
|
Mass-spring problems.
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|
|
13, 14, 17, 20, 24, 26
|
Mass-spring problems - parameter exploration.
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|
|
15
|
Superposition of IC solutions.
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|
|
27
|
An oscillating floating object.
|
| 3.8
|
1-4
|
Sum of trig functions with different period (beats).
|
|
|
5-8
|
Set up (5,6) and solve (7,8) an IVP given a description of the physical system (forced mass-spring).
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|
|
9, 10, 11, 17, 18
|
Forced mass-spring problems.
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Chapter 6
| Section
|
Problems
|
Description
|
| 6.1
|
1-4
|
Graph piecewise defined functions.
|
|
|
5-6
|
These problems expect you to find the Laplace transforms from the definition (not from a table).
|
|
|
21-24
|
Find transforms of piecewise defined functions.
|
| 6.2
|
1-10
|
Find inverse transforms (rational functions, mostly with quadratic denominators).
|
|
|
11-23
|
Solve IVPs using Laplace transforms.
|
|
|
24-27
|
Find transforms of ODE solution when forcing is a piecewise defined function.
|
|
|
29-35
|
Transforming functions of the form $g(t)=tf(t)$ when the transform of $f(t)$ is known.
|
| 6.3
|
1-6
|
Sketch functions expressed in terms of the Heaviside function.
|
|
|
7-12
|
Express piecewise functions in terms of Heavisides.
|
|
|
13-18
|
Transform piecewise functions.
|
|
|
19-24
|
Invert transformed functions.
|
|
|
30-33
|
|
| 6.4
|
|
To appear soon.
|
| 6.5
|
|
To appear soon.
|
| 6.6
|
|
To appear soon.
|
Chapter 7
| Section
|
Problems
|
Description
|
| 7.1
|
1-13
|
You should be able to do these but their purpose is mostly to introduce the connection between 2x2 systems and second order ODEs and will not appear on the midterm/final so don't spend much time here.
|
| 7.2
|
1-21
|
Only do these if you need to review the basics of matrix algebra.
|
|
|
22-24
|
Check that the given vector functions satisfy the given systems of equations.
|
| 7.3
|
1-15
|
Only do these if you need to review the basics of matrix algebra.
|
|
|
16-21
|
Practice finding eigenvalues and eigenvectors.
|
| 7.5
|
1-6
|
Solve 2x2 system of ODEs, describe $t\to\infty$ behaviour, draw direction field and some solutions in the phase plane.
|
|
|
7-8
|
Worth thinking about but we didn't talk about this case explicitly (solving yes, but sketching no).
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|
|
24-27
|
Sketching trajectories in the phase plane and as functions of $t$.
|
|
|
29, 30, 31
|
For 31, instead of finding eigenvalues, just calculate trace and det of the matrix and use those to classify the steady state (equilibrium) as stable node, stable spiral, unstable spiral etc. and find the transition value of $\alpha$.
|
| 7.6
|
1-6, 9-12.
|
Solve system, describe long-term behaviour. For 11 and 12, omit (d).
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|
|
13-20
|
2x2 systems with a parameter - find value of parameter at which solution change from inward to outward spirals (or vice versa).
|
|
|
28
|
Mass-spring (second order) equation converted to first order 2x2 system.
|
| 7.8
|
1-4, 7-10
|
Solve system, describe long-term behaviour.
|
|
|
16
|
LRC circuit with repeated root.
|
| 7.9
|
22, 23 from 7.1
|
Two-tank mixing problem; for both of these, also solve the system of equations from part (a).
|
|
|
|
Invent your own two-tank problems by modifying Figure 7.1.6 (problem 22 from 7.1) so that there are different inflow, outflow and cross-flow-between-tank arrangements.
|
Chapter 10
| Section
|
Problems
|
Description
|
| 10.2
|
1-8
|
Determine if function is periodic; if so, find its fundamental period.
|
|
|
13-18
|
Sketch graph of given piecewise constant/linear function for three periods. Find Fourier series.
|
|
|
19-24
|
Sketch graph of given piecewise continuous function for three periods. Find Fourier series. Plot partial sums. Describe convergence. I suggest you use Wolfram Alpha for some of the messier integrals and Desmos.com or similar for the plotting.
|
| 10.3
|
1-6
|
Find Fourier series for piecewise constant/linear function on $[0,L]$ extended periodically. Sketch the graph of the series. Note - the extension here is neither even nor odd, just periodic with period L so the Fourier series will have both sine and cosine terms.
|
|
|
13-16
|
Using Fourier series to solve second order ODE with piecewise forcing (could also use Laplace transforms for these - see Section 6.3 Problems 34-38).
|
| 10.4
|
1-6
|
Is it even, odd or neither?
|
|
|
7-12
|
Sketch even/odd extensions of given function.
|
|
|
15-22
|
Find the Fourier sine/cosine series.
|
|
|
35-36
|
Interesting applications of Fourier series to estimating $\pi$.
|
|
|
39
|
Extending $f$ about both $x=0$ and $x=L$ to get an alternate Fourier series.
|
| 10.5
|
7-12
|
Solve Diffusion equation with Dirichlet BCs
|
| 10.6
|
1-8
|
Find the steady state (particular) solution to the given BCs.
|
|
|
9-17
|
Various Heat conduction problems with nonhomogeneous and/or mixed Dirichlet/Neumann BCs.
|
