Difference between revisions of "Suggested problems"
From UBCMATH WIKI
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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. | 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=== | ||
{|style="border-collapse: collapse;" | {|style="border-collapse: collapse;" | ||
| − | !Section | + | !style="width: 75px;"|Section |
| − | ! | + | !style="width: 100px;"|Problems |
| − | ! Description | + | !Description |
|- | |- | ||
!style="border-style: solid; border-width: 1px 0 0 0"|1.1-1.2 | !style="border-style: solid; border-width: 1px 0 0 0"|1.1-1.2 | ||
| Line 14: | Line 15: | ||
|style="border-style: solid; border-width: 1px 0 0 0"|Classify by order, linearity | |style="border-style: solid; border-width: 1px 0 0 0"|Classify by order, linearity | ||
|- | |- | ||
| − | + | ! | |
|7-14 | |7-14 | ||
|Verify that a given function is a solution to particular ODE. | |Verify that a given function is a solution to particular ODE. | ||
|- | |- | ||
| − | + | ! | |
|19-20 | |19-20 | ||
|Find the value of $r$ so that $t^r$ solves the given ODE. | |Find the value of $r$ so that $t^r$ solves the given ODE. | ||
| + | |- | ||
| + | |} | ||
| + | |||
| + | ===Chapter 2=== | ||
| + | {|style="border-collapse: collapse;" | ||
| + | !style="width: 75px;"|Section | ||
| + | !style="width: 100px;"|Problems | ||
| + | !Description | ||
|- | |- | ||
!style="border-style: solid; border-width: 1px 0 0 0"|2.1 | !style="border-style: solid; border-width: 1px 0 0 0"|2.1 | ||
| Line 26: | Line 35: | ||
|style="border-style: solid; border-width: 1px 0 0 0"|Problems which can be solved using an integrating factor. Try using Method of Undetermined Coefficients as well for practice. | |style="border-style: solid; border-width: 1px 0 0 0"|Problems which can be solved using an integrating factor. Try using Method of Undetermined Coefficients as well for practice. | ||
|- | |- | ||
| − | + | ! | |
|30 | |30 | ||
|Choosing ICs to ensure boundedness of solutions | |Choosing ICs to ensure boundedness of solutions | ||
|- | |- | ||
| − | + | ! | |
|34-37 | |34-37 | ||
|Construct equations whose solutions have the prescribed asymptotic behaviour. | |Construct equations whose solutions have the prescribed asymptotic behaviour. | ||
| Line 42: | Line 51: | ||
|style="border-style: solid; border-width: 1px 0 0 0"|Salt water inflow/outflow modeling problems | |style="border-style: solid; border-width: 1px 0 0 0"|Salt water inflow/outflow modeling problems | ||
|- | |- | ||
| − | + | ! | |
|17 | |17 | ||
|Stephan-Boltzmann heat loss. | |Stephan-Boltzmann heat loss. | ||
|- | |- | ||
| − | + | ! | |
|18 | |18 | ||
|First order forcing (temperature). | |First order forcing (temperature). | ||
| + | |- | ||
| + | |} | ||
| + | |||
| + | ===Chapter 3=== | ||
| + | |||
| + | {|style="border-collapse: collapse;" | ||
| + | !style="width: 75px;"|Section | ||
| + | !style="width: 100px;"|Problems | ||
| + | !Description | ||
|- | |- | ||
!style="border-style: solid; border-width: 1px 0 0 0"|3.1 | !style="border-style: solid; border-width: 1px 0 0 0"|3.1 | ||
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|style="border-style: solid; border-width: 1px 0 0 0"| | |style="border-style: solid; border-width: 1px 0 0 0"| | ||
|- | |- | ||
| − | + | ! | |
|9-16 | |9-16 | ||
| | | | ||
|- | |- | ||
| − | + | ! | |
|17-18 | |17-18 | ||
| | | | ||
|- | |- | ||
| − | + | ! | |
|21-24 | |21-24 | ||
| | | | ||
|- | |- | ||
| − | + | ! | |
|28 | |28 | ||
| | | | ||
| Line 74: | Line 92: | ||
|style="border-style: solid; border-width: 1px 0 0 0"| | |style="border-style: solid; border-width: 1px 0 0 0"| | ||
|- | |- | ||
| − | + | ! | |
|24-27 | |24-27 | ||
| | | | ||
|- | |- | ||
| − | + | ! | |
|28 | |28 | ||
| | | | ||
| Line 86: | Line 104: | ||
|style="border-style: solid; border-width: 1px 0 0 0"| | |style="border-style: solid; border-width: 1px 0 0 0"| | ||
|- | |- | ||
| − | + | ! | |
|27 | |27 | ||
| | | | ||
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|style="border-style: solid; border-width: 1px 0 0 0"| | |style="border-style: solid; border-width: 1px 0 0 0"| | ||
|- | |- | ||
| − | + | ! | |
|23-30 | |23-30 | ||
|Reduction of order - omit this year (2014) | |Reduction of order - omit this year (2014) | ||
| Line 102: | Line 120: | ||
|style="border-style: solid; border-width: 1px 0 0 0"| | |style="border-style: solid; border-width: 1px 0 0 0"| | ||
|- | |- | ||
| − | + | ! | |
|30 | |30 | ||
| | | | ||
| Line 110: | Line 128: | ||
|style="border-style: solid; border-width: 1px 0 0 0"|Converting from sum of sin and cos to a single cos expression. | |style="border-style: solid; border-width: 1px 0 0 0"|Converting from sum of sin and cos to a single cos expression. | ||
|- | |- | ||
| − | + | ! | |
|6, 7, 11 | |6, 7, 11 | ||
|Mass-spring problems. | |Mass-spring problems. | ||
|- | |- | ||
| − | + | ! | |
|13, 14, 17, 20, 24, 26 | |13, 14, 17, 20, 24, 26 | ||
|Mass-spring problems - parameter exploration. | |Mass-spring problems - parameter exploration. | ||
|- | |- | ||
| − | + | ! | |
|15 | |15 | ||
|Superposition of IC solutions. | |Superposition of IC solutions. | ||
|- | |- | ||
| − | + | ! | |
|27 | |27 | ||
|An oscillating floating object. | |An oscillating floating object. | ||
| Line 130: | Line 148: | ||
|style="border-style: solid; border-width: 1px 0 0 0"|Sum of trig functions with different period (beats). | |style="border-style: solid; border-width: 1px 0 0 0"|Sum of trig functions with different period (beats). | ||
|- | |- | ||
| − | + | ! | |
|5-8 | |5-8 | ||
|Set up (5,6) and solve (7,8) an IVP given a description of the physical system (forced mass-spring). | |Set up (5,6) and solve (7,8) an IVP given a description of the physical system (forced mass-spring). | ||
|- | |- | ||
| − | + | ! | |
|9, 10, 11, 17, 18 | |9, 10, 11, 17, 18 | ||
|Forced mass-spring problems. | |Forced mass-spring problems. | ||
| + | |- | ||
| + | |} | ||
| + | |||
| + | ===Chapter 6=== | ||
| + | |||
| + | {|style="border-collapse: collapse;" | ||
| + | !style="width: 75px;"|Section | ||
| + | !style="width: 100px;"|Problems | ||
| + | !Description | ||
|- | |- | ||
!style="border-style: solid; border-width: 1px 0 0 0"|6.1 | !style="border-style: solid; border-width: 1px 0 0 0"|6.1 | ||
| − | |style="border-style: solid; border-width: 1px 0 0 0"| | + | |style="border-style: solid; border-width: 1px 0 0 0"|1-4 |
| − | |style="border-style: solid; border-width: 1px 0 0 0"| | + | |style="border-style: solid; border-width: 1px 0 0 0"|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. | ||
|- | |- | ||
!style="border-style: solid; border-width: 1px 0 0 0"|6.2 | !style="border-style: solid; border-width: 1px 0 0 0"|6.2 | ||
| − | |style="border-style: solid; border-width: 1px 0 0 0"| | + | |style="border-style: solid; border-width: 1px 0 0 0"|1-10 |
| − | |style="border-style: solid; border-width: 1px 0 0 0"| | + | |style="border-style: solid; border-width: 1px 0 0 0"|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. | ||
|- | |- | ||
!style="border-style: solid; border-width: 1px 0 0 0"|6.3 | !style="border-style: solid; border-width: 1px 0 0 0"|6.3 | ||
| Line 161: | Line 208: | ||
|style="border-style: solid; border-width: 1px 0 0 0"| | |style="border-style: solid; border-width: 1px 0 0 0"| | ||
|style="border-style: solid; border-width: 1px 0 0 0"|To appear soon. | |style="border-style: solid; border-width: 1px 0 0 0"|To appear soon. | ||
| + | |- | ||
| + | |} | ||
| + | |||
| + | ===Chapter 7=== | ||
| + | |||
| + | {|style="border-collapse: collapse;" | ||
| + | !style="width: 75px;"|Section | ||
| + | !style="width: 100px;"|Problems | ||
| + | !Description | ||
|- | |- | ||
!style="border-style: solid; border-width: 1px 0 0 0"|7.1 | !style="border-style: solid; border-width: 1px 0 0 0"|7.1 | ||
| Line 170: | Line 226: | ||
|style="border-style: solid; border-width: 1px 0 0 0"|Only do these if you need to review the basics of matrix algebra. | |style="border-style: solid; border-width: 1px 0 0 0"|Only do these if you need to review the basics of matrix algebra. | ||
|- | |- | ||
| − | + | ! | |
|22-24 | |22-24 | ||
|Check that the given vector functions satisfy the given systems of equations. | |Check that the given vector functions satisfy the given systems of equations. | ||
| Line 178: | Line 234: | ||
|style="border-style: solid; border-width: 1px 0 0 0"|Only do these if you need to review the basics of matrix algebra. | |style="border-style: solid; border-width: 1px 0 0 0"|Only do these if you need to review the basics of matrix algebra. | ||
|- | |- | ||
| − | + | ! | |
|16-21 | |16-21 | ||
|Practice finding eigenvalues and eigenvectors. | |Practice finding eigenvalues and eigenvectors. | ||
| Line 186: | Line 242: | ||
|style="border-style: solid; border-width: 1px 0 0 0"|Solve 2x2 system of ODEs, describe $t\to\infty$ behaviour, draw direction field and some solutions in the phase plane. | |style="border-style: solid; border-width: 1px 0 0 0"|Solve 2x2 system of ODEs, describe $t\to\infty$ behaviour, draw direction field and some solutions in the phase plane. | ||
|- | |- | ||
| − | + | ! | |
|7-8 | |7-8 | ||
|Worth thinking about but we didn't talk about this case explicitly (solving yes, but sketching no). | |Worth thinking about but we didn't talk about this case explicitly (solving yes, but sketching no). | ||
|- | |- | ||
| − | + | ! | |
|24-27 | |24-27 | ||
|Sketching trajectories in the phase plane and as functions of $t$. | |Sketching trajectories in the phase plane and as functions of $t$. | ||
|- | |- | ||
| − | + | ! | |
|29, 30, 31 | |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 $\alpha$. | |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 $\alpha$. | ||
| Line 202: | Line 258: | ||
|style="border-style: solid; border-width: 1px 0 0 0"|Solve system, describe long-term behaviour. For 11 and 12, omit (d). | |style="border-style: solid; border-width: 1px 0 0 0"|Solve system, describe long-term behaviour. For 11 and 12, omit (d). | ||
|- | |- | ||
| − | + | ! | |
|13-20 | |13-20 | ||
|2x2 systems with a parameter - find value of parameter at which solution change from inward to outward spirals (or vice versa). | |2x2 systems with a parameter - find value of parameter at which solution change from inward to outward spirals (or vice versa). | ||
|- | |- | ||
| − | + | ! | |
|28 | |28 | ||
|Mass-spring (second order) equation converted to first order 2x2 system. | |Mass-spring (second order) equation converted to first order 2x2 system. | ||
| Line 214: | Line 270: | ||
|style="border-style: solid; border-width: 1px 0 0 0"|Solve system, describe long-term behaviour. | |style="border-style: solid; border-width: 1px 0 0 0"|Solve system, describe long-term behaviour. | ||
|- | |- | ||
| − | + | ! | |
|16 | |16 | ||
|LRC circuit with repeated root. | |LRC circuit with repeated root. | ||
| Line 222: | Line 278: | ||
|style="border-style: solid; border-width: 1px 0 0 0"|Two-tank mixing problem; for both of these, also solve the system of equations from part (a). | |style="border-style: solid; border-width: 1px 0 0 0"|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. | |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. | ||
|- | |- | ||
|} | |} | ||
Revision as of 15:20, 15 March 2014
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.
Contents |
Chapter 1
| Section | Problems | Description |
|---|---|---|
| 1.1-1.2 | 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 |
| 7-14 | Verify that a given function is a solution to particular ODE. | |
| 19-20 | Find the value of $r$ so that $t^r$ solves the given ODE. |
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. |
| 30 | Choosing ICs to ensure boundedness of solutions | |
| 34-37 | Construct equations whose solutions have the prescribed asymptotic behaviour. | |
| 2.2 | 1-8 | Separable equations |
| 2.3 | 1-4 | Salt water inflow/outflow modeling problems |
| 17 | Stephan-Boltzmann heat loss. | |
| 18 | First order forcing (temperature). |
Chapter 3
| Section | Problems | Description |
|---|---|---|
| 3.1 | 1-8 | |
| 9-16 | ||
| 17-18 | ||
| 21-24 | ||
| 28 | ||
| 3.2 | 1-6 | |
| 24-27 | ||
| 28 | ||
| 3.3 | 7-22 | |
| 27 | ||
| 3.4 | 1-14 | |
| 23-30 | Reduction of order - omit this year (2014) | |
| 3.5 | 1-20 | |
| 30 | ||
| 3.7 | 1-4, 16 | Converting from sum of sin and cos to a single cos expression. |
| 6, 7, 11 | Mass-spring problems. | |
| 13, 14, 17, 20, 24, 26 | Mass-spring problems - parameter exploration. | |
| 15 | Superposition of IC solutions. | |
| 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). | |
| 9, 10, 11, 17, 18 | Forced mass-spring problems. |
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 | To appear soon. | |
| 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). | |
| 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). |
| 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. |
