Module 1:  Introduction; First-Order Linear Equations; First-Order Nonlinear Equations

Assignments 1 - 13

Test 1 assigned as part of Assignment 14.

Asst Number, original date of creation during Spring Semester 2011

qa

Class Notes / QA Topics

Associated Text Section(s)

First Edition

Query

Query Topic (note Query topic often runs 2 or even 3 assignments 'behind' qa topic)

class notes reference

video link

Preliminary

You should be working on the Orientation and Startup for your course.  The initial assignments should be completed within the first few days of the term. 

You will learn how to complete or submit assignments when you complete the Orientation and Startup.

initial videos

#0

qa 00 

• qa_00 is an assessment/review of some basic first-year calculus topics
• the document is in qa format but, so that it may function as an assessment tool, the document does not contain given solutions

qa 00 

#1:  View initial videos and take notes.

Begin work on qa_01

See

Spring 2012

link for Spring 2012 Class Notes

This assignment includes sufficient introductory material to be done before receiving the textbook.

• m x '' = f(t, x, x') and some solns
• basic terminology
• approx solns
• direction fields

brief synopsis each asst

Ch 1

Recommended:  Begin work on qa 01

Partial submission in permissible.

110110

See also

Spring 2012

1.      Identify the order of a given differential equation and the orders of the derivatives present in the equation.

2.      Write the form of a given differential equation.

3.      Apply basic terminology in describing differential equations and their solutions.

4.      Determine whether a given function constitutes a solution to a given differential equation.

5.      Given a first-order differential equation in the form x ‘ = f(t, x), an initial condition x_0 = x(t_0) and a t-interval `dt find the approximate value of x(t_0 + `dt); iterate this process as specified.

6.      Given a first-order differential equation in the form x ‘ = f(t, x), construct a direction field over a specified region and use the direction field to sketch a family of approximate solution curves.

7.      Given a direction field for a first-order differential equation over a specified region, use the direction field to sketch a family of approximate solution curves.

#2

qa 01

• problems related to introduction

Ch 1

(query 00)

• existence and uniqueness

110119

disk1

1.      Given a differential equation identify intervals over which solutions can be guaranteed to exist.

2.      In terms of the behavior of the direction field of a given equation discuss on which intervals solutions can be expected to exist.

#3

qa 02

• first-order linear homogeneous
• first-order linear nonhomogeneous

2.1, 2.2

Section 2.2 Problems

Query 01

• first-order linear homogeneous

110124

1.      Given a first-order linear homogeneous equation and an initial condition find and verify its solution.

2.      Solve applications problems by writing and solving appropriate first-order differential equations and interpreting the solutions.

#4

qa 03

• dP/dt = k P; dP/dt = k P + m as lin eqns
• first mixing problem

2.2 continued

Section 2.3 Problems

Query 02

• first-order linear non-homogeneous

110131

1.       Given a first-order linear homogeneous equation and an initial condition find and verify its solution.

2.      Solve applications problems by writing and solving appropriate first-order differential equations and interpreting the solutions.

3.      Use differential equations to solve basic mixing problems.

#5

qa 04

• direction field; scaling of direction field
• temp relaxation as lin equations
• separable equations
• implicit differentiation, exact equations

2.3

Section 2.4 Problems

Query 03

• exp growth/decay
• growth with migration

110202

1.      Apply appropriate techniques to scale direction fields when the scales of the two coordinate axes differ.

2.      Solve problems involving temperature relaxation, in which the rate of change of the temperature of an object is proportional to the difference between its temperature and that of its surroundings.

3.      Identify and solve separable differential equations with initial conditions.

4.      Identify situations in which are characterized by a constant (positive or negative) growth rate; set up and solve equations to model such situations.

#6

qa 05

• effect of vertical asymptote in direction field
• form of equation
• exact eqns as differentials
• test for exactness

2.4

Section 2.5 Problems

Query 04

• mixing
• temperature relaxation

110207

1.      Describe the possible effects on solution curves of a vertical asymptote in a direction field.

2.      Verify that if the total differential of a function of two variables is set equal to a constant, the result is a first-order differential equation.

3.      Test a given differential equation for exactness.

4.      Given a differential equation verified to be exact, solve it.

5.      Given a mixing problem which can be modeled with a first-order differential equation, write the equation and solve it.

#7

qa 06

• existence
• air resistance quadratic in v
• exact
• bernoulli intro
• logistic intro

2.5, 2.6

Section 3.1, 3.2 Problems

Query 05

• separable(?) equations
• match graphs of solutions with equations

110209

1.      Match graphs of solution curves with equations.

2.      Set up and solve first-order equations for the velocity of constant mass under the influence of a constant force in addition to a force proportional to the first or second power of the velocity.

3.      Use the transformation v = y^m to transform a Bernoulli equation into a linear differential equation, solve the equation and verify your solution.

4.      Solve the logistic equation by separation of variables and partial fractions.

#8

qa 07

• logistic equation etc
• linear, quadratic drag force
• dv/dt = dv/dx * dx/dt etc

2.7

Section 3.3 Problems

Query 06

• exact equations

110214

1.      Given a situation involving logistic growth, set up and solve the appropriate equation and interpret the solutions.

2.      Given a situation in which the net force on a constant mass is a function of position use the chain rule dv / dt = dv/dx * dx/dt to change the dependent variable from t to x; solve and interpret the resulting equation.

#9

qa 08

• euler method
• overestimate or underestimate
• effect of step size
• airsoft BB shot

2.8

Section 3.4 Problems

Query 07

• Bernoulli equations

110216

1.      Apply the Euler method to obtain an approximate solution curve for a given first-order differential equation, and place appropriate limits on the accuracy your solution.

2.      Determine whether a result obtained from the Euler method is an overestimate or an underestimate, and place appropriate limits on the extent of the error.

3.      Discuss the effect of step size on the accuracy of an Euler approximation.

4.      Solve Bernoulli equations and interpret the solutions.

#10

qa 09

• second order sub y = e^(rt)
• Wronskian
• span of set of functions
• Euler identity
• linear independence

2.8 continued

Section 3.5 Problems

Query 08

• logistic

110221

1.      Find solutions of second-order homogeneous equations a y ‘’ + b y + c = 0 with constant coefficients by substituting y = e^(r t).

2.      Know the Euler identity and apply it to solutions in cases where r values are imaginary or complex.

3.      Evaluate the Wronskian of set of functions to determine whether it is linearly independent.

4.      Describe the span of a given set of functions.

#11

qa 10

• more second order
• region of definition 2d order nonconst coeff
• more Wronskian
• where Wronskian comes from
• implications if y = u e^(r t)
• two lin ind solns for simple pend

2.9

Section 3.6 Problems

Query 09

• drag
• vertical projectiles

110223

1.      Determine the intervals on which continuous solutions to a given second-order equation with nonconstant coefficients would be expected to exist.

2.      Show what is obtained if y = u e^(r t) is substituted into a homogeneous linear equation, where r is a solution to the characteristic equation.

#12

qa 11

• repeated root: y = u e^(r t) etc.
• LRC homogeneous
• LRC nonhomogeneous
• LRC analogous to spring with air drag

2.9 continued

Section 3.7 Problems

Query 10

• resisted motion

110228

1.      Solve second-order linear constant coefficient equations for which the characteristic equation yields a repeated root.

2.      Solve a situation involving an LRC circuit for the case where a voltage source is not present.

3.  Analyze a situation involving motion of an object subject to the force of a spring in which a drag force proportional to velocity is also present.

#13

qa 12

• undetermined coefficients
• try y = u1 y1 + u2 y1 etc (var param)
• then with y_1 u_1 ' + y_2 u_2 ' = 0

2.10

Section 3.8 Problems

Query 11

• Euler's method

110302

1.      Solve nonhomogeneous second-order linear equations with constant coefficients using the method of undetermined coefficients.

2.      Solve nonhomogeneous second-order linear equations with constant coefficients using the method of variation of parameters.

3.      Apply Euler’s method to second-order equations.

Module 2:  Second-Order Equations

Assignments 14 - 23

#14

qa 13

• interpret same eqn as damped pend, LRC

3.1

Section 4.1 Problems

Query 12

• floating cylinder
• intervals of solution
• nature of solution

110307

 Complete Test 1 within about a week of completing Assignment 13

1.      Interpret a given second-order equation with constant coefficients in terms of the stored charge in an LRC circuit, and in terms of the motion of a damped pendulum.

2.  Write the differential equation for a cylinder of given dimensions and density bobbing in a fluid of given density.

3.  Given a second-order linear differential equation with initial condition, determine the maximum interval over which a solution can be guaranteed to exist.

4.  Given a second-order differential equation and the slope at a given point in the plane, determine the concavity of the solution curve in the vicinity of that point.

#15

qa 14

• 2d order eqns anticipating applications, interp

3.2

Section 4.2 Problems

Query 13

(largely but not wholly redundant with 12)

10309

1.  Solve second-order linear equations with constant coefficients, for which the coefficient of the first derivative is zero.

2.  Verify whether a given set of solutions constitutes a fundamental set on a given interval.

#16

qa 16

• systems of equations
• systems and matrices
• eigenvalue problem

3.2 continued

Section 4.3 Problems

Query 14

• lin ind solutions
• Wronskian

110321

 

1.  Express a given system of linear differential equations as a matrix equation.

2.  Calculate the derivative and/or an antiderivative of a given matrix.

3.  Solve the eigenvalue problem for a given matrix.

4.  Verify whether a given set of solutions to a differential equation is linearly independent.

#17

qa 17

from 0323; not yet in qa form?

Query 15

• fund sets
• Wronskians

(0323 saved over with physics; locate photos)

#18

qa 18

• qualitative rabbits and wolves
• impossible problem posed (should be fixed)
• mixing problem

3.3

Section 4.4 Problems

Query 16

• solns to equations
• solns as t -> infinity

110328

 

1.  Write the system of equations for and describe the qualitative behavior of the rabbits-and-wolves model.

2.  Write and solve the system of linear equations for a two-tank mixing problem with constant flows.

3.  Solve second-order constant coefficient equations.

4.  Solve problems involving mathematical and real-world applications of second-order constant coefficient equations.

#19

qa 19

• rabbits and wolves, phase plane
• two-spring system
• RC, response to sine-wave voltage
• LRC, response to sine-wave voltage
• Laplace Transform

3.4, 3.5

Section 4.5, 4.6 Problems

Query 17

• solve equations
• solve nonconst coeff given one soln

110330

 

1.  Analyze the phase plane to describe the general behavior of solutions of the rabbits-and-wolves model.

2.  Write and solve second-order constant coefficient equations to model two-spring systems, RC circuits and LRC circuits.

3.  For various functions f(t) integrate f(t) * e^(-2 t) from t = 0 to infinity.

4.  Solve second-order differential equations for which the characteristic equation has repeated real roots or complex conjugate roots.

5.  Apply the method of reduction of order to find a second solution of the second-order linear equation given one solution.

#20

qa 20

• higher order eqn as system
• mixing problem

3.6

Section 4.7 Problems

Query 18

• mass on spring
• spring-dashpot system

110404

 

1.  Express a third- or higher-order system of linear differential equations with constant coefficients as a system of linear differential equations.

2.  Given a physical system which undergoes linearly damped or free unforced harmonic motion, write the second-order equation (wht appropriate initial conditions) which models its motion, solve the equation, and interpret the solution.

#21

qa 21

• more Laplace Transforms
• find fn from transform
• mixing problem with slight modification

3.7, 3.8

Section 4.8, 4.9 Problems

Query 19

• undetermined coefficients

110406

 

1.  Using the definition find the Laplace Transform of a given function.

2.  Using the definition express the Laplace Transform of the first derivative of a given function in terms of the Laplace transform of the function itself and an initial condition.

3.  Show that the Laplace Transform is a linear operator.

4.  Correctly identify the specific Laplace Transforms of a function whose transform you have previously derived, and correctly identify the specific function associated with the given transform.

5.  Apply the method of undetermined coefficients, where it is possible to do so, to find a particular solution of a given second-order nonhomogeneous equation, and combine with the solution of the homogeneous equation to obtain a general solution.

#22

qa 22

• Abel's Theorem, start
• Laplace to solve simple 1st order eqn (exp xc
• Heaviside fn
• beginning square wave
• more properties of Laplace transform

3.9

Section 4.10 Problems

Query 20

• variation of parameters

110411

 

1.  Apply Laplace Transforms to solve the equation y ' = - k y.

2.  Using the definition of Laplace Transform, find the transform of the Heaviside function h(t), and of h(t - alpha).

3.  Use the Heaviside function to construct a square-wave function, and find its Laplace Transform.

4.  Use the definition of the Laplace Transform to find the transform of the second derivative of a function in terms of the transform of its first derivative, then in terms of the transform of the function itself.

5.  Apply the method of variation of parameters to find the solution of a given second-order linear nonhomogeneous equation, given the solutions of the associated homogeneous equation.

#23

qa 23

• verify Abel's theorem for specific example
• geometric series and Laplace Transform
• more Laplace Transforms

3.10

Section 4.11 Problems

Query 21

• driven systems

110413

 

1.   Analyze mechanical and electrical systems which can be modeled by homogeneous and nonhomeogeneous second-order linear differential equations with constant coefficients; model a given system, solve the equation, apply initial conditions and interpret the solution.

Module 3:  Systems of Linear Equations and Laplace Transforms

Assignments 24 - 34

#24

qa 24

• solve system, use Wronskian
• Laplace Tranform of periodic function

4.1

Section 6.1 Problems

Query 22

• properties of matrices
• matrices and systems of diff eq
• vector form of system

110418

 Complete Test 2 within about a week of completing Assignment 23

 

1.   Solve a linear system of two equations with constant coefficients for which only one independent solution results from the solution to the eigenvalue problem.

2.  Apply the periodicity property of the Laplace Transform to solve a nonhomogeneous second-order equation with constant coefficients which models a physical system driven by a square wave.

3.  Represent a linear system of differential equations as the product of a matrix, a column vector representing the solution and the derivative of that column vector.

4.  Apply calculus to matrix functions and products of matrix functions.

#25

qa 25

• Dirac delta
• solve system

3.2

Section 6.2 Problems

Query 23

• intervals of solution
• system represents what higher-order eqn

110420

 

1.  For a given system of linear equations with initial conditions, find the largest interval over which a solution is guaranteed.

2.  For a given system of two or more first-degree linear equations find a higher-order equation whose solutions are equivalent.

3.  Apply the Wronskian to determine whether a given set of solutions of a linear system is a fundamental set.

#26

qa 26

• psi matrix
• change of basis

4.3

Section 6.3 Problems

Query 24

• verify Abel's Theorem
• psi matrix

110425

 

1.  Apply Abel's Theorem to find the Wronskian of a given system for a given value of the dependent variable, given the value of the Wronskian at another given value of that variable.

2.  Verify whether a given set of solutions to a homogeneous system of linear differential equations is linearly independent.

3.  Show that the fundamental matrix (the psi matrix) for a given system can itself be interpreted as a solution of the system.

4.  Determine whether the product of the fundamental matrix for a given system, with another given matrix, results in a fundamental matrix.

5.  Find a fundamental matrix for a given system, and for a given initial condition on the fundamental matrix, use matrix operations to find the solution which satisfies the given condition.

#27

qa 27

• transformation expressed in eigenvector basis

4.4, 4.5

Section 6.4, 6.5 Problems

Query 25

• properties of psi matrix

110427

 

1.  Verify whether a given set of solutions to a homogeneous system of linear differential equations is linearly independent.

2.  Show that the fundamental matrix (the psi matrix) for a given system can itself be interpreted as a solution of the system.

3.  Determine whether the product of the fundamental matrix for a given system, with another given matrix, results in a fundamental matrix.

4.  Find a fundamental matrix for a given system, and for a given initial condition on the fundamental matrix, use matrix operations to find the solution which satisfies the given condition.

#28

qa 28

• propagator matrix
• exponential of matrix
• variation of parameters for systems

 4.6

Section 6.6 Problems

Query 26

• solve system
• solve mixing equation

110502

 

1.  Solve linear systems of constant-coefficient differential equations having distinct real eigenvalues.

2.  Solve linear systems of constant-coefficient differential equations having complex conjugate eigenvalues.

3.  Solve mixing problems involving two or more tanks and constant flow rates.

4.  Solve vector differential equations.

#29

qa 29

• summary of psi matrix, propagator, diagonalization

4.7, 4.8

Section 6.7, 6.8 Problems

Query 27

• solve system to find eigenvalues, eigenvectors
• find particular solution
• apply to motion in force field

110504

 

1.  Solve linear systems of constant-coefficient differential equations having repeated eigenvalues.

2.  Use variation of parameters to solve nonhomogeneous linear systems.

#30

4.9

Section 6.10 Problems

Query 28

• diagonalize
• find algebraic, geometric multiplicity
• solve systems using above

 

1.  Given an n x n matrix determine whether it has n linearly independent eigenvectors.

2.  Given an n x n matrix with n linearly independent eigenvectors, use a change-of-basis matrix to perform the similarity transformation required to diagonalize it.

3.  Given a first-order linear system of differential equations with constant coefficients determine if the matrix associated with the system is diagonalizable.

3.  Given a first-order linear system of differential equations with constant coefficients, representable by a diagonalizable matrix, use the appropriate change-of-basis matrix to diagonalize the system and uncouple the equations, solve, and use the inverse of the change-of-basis matrix to transform the solution back into the original basis, and interpret to obtain the solution of the original system.

#31

Section 6.11 Problems

Query 29

• propagator matrix
• functions of a matrix

 

1.  Given a linear system of differential equations, find the fundamental matrix and use it to find the propagator matrix phi(t, s) = psi(t) psi^-1(s).

2.  Find the exponential, sine, cosine or square root of a diagonal matrix.

3.  Given a matrix and an associated full set of eigenpairs (eigenvalues and eigenvectors), ...

... matrices don't generally commute with their derivatives so we can't push the analogy too far ...

#32

5.1

Section 7.1 Problems

Query 30

• find Laplace Transform for given fns using definition

 

1.  Using the definition of the Laplace Transform find the transform and its domain for a given function.

#33

5.2, 5.3

Section 7.2, 7.3 Problems

Query 31

• using table if needed find Laplace Transform of given fn

 

1.  Use a table of Laplace Transforms to find the transforms of given functions, and/or to find inverse transforms of given expressions.

2.  Graph given combinations of Heaviside functions and determine their transforms.

3.  Combine multiples and shifts of Heaviside functions to obtain given periodic waveforms including but not limited to square and sawtooth patterns.

4.  Apply shift theorems

5.  Apply the method of partial fractions to obtain inverse Laplace Transforms.

6.  Solve initial given value problems using Laplace Transforms.

#34

5.4

Section 7.4 Problems

Query 32

• use Laplace Transform to solve given eqn, including periodic fns

Complete Test 3 by the end of the final exam period 

 

1.  Use Laplace Transforms to solve differential equations representing systems with periodic driving functions.