# 4.6: The Method of Undetermined Coefficients I - Mathematics

In this section we consider the constant coefficient equation

[label{eq:5.4.1} ay''+by'+cy=e^{alpha x}G(x),]

where (alpha) is a constant and (G) is a polynomial.

From Theorem 5.3.2, the general solution of Equation ef{eq:5.4.1} is (y=y_p+c_1y_1+c_2y_2), where (y_p) is a particular solution of Equation ef{eq:5.4.1} and ({y_1,y_2}) is a fundamental set of solutions of the complementary equation

[ay''+by'+cy=0. onumber ]

In Section 5.2 we showed how to find ({y_1,y_2}). In this section we’ll show how to find (y_p). The procedure that we’ll use is called the method of undetermined coefficients. Our first example is similar to Exercises 5.3.16-5.3.21.

Example (PageIndex{1}):

Find a particular solution of

[label{eq:5.4.2} y''-7y'+12y=4e^{2x}.]

Then find the general solution.

Solution

Substituting (y_p=Ae^{2x}) for (y) in Equation ef{eq:5.4.2} will produce a constant multiple of (Ae^{2x}) on the left side of Equation ef{eq:5.4.2}, so it may be possible to choose (A) so that (y_p) is a solution of Equation ef{eq:5.4.2}. Let’s try it; if (y_p=Ae^{2x}) then

[y_p''-7y_p'+12y_p=4Ae^{2x}-14Ae^{2x}+12Ae^{2x}=2Ae^{2x}=4e^{2x} onumber]

if (A=2). Therefore (y_p=2e^{2x}) is a particular solution of Equation ef{eq:5.4.2}. To find the general solution, we note that the characteristic polynomial of the complementary equation

[label{eq:5.4.3} y''-7y'+12y=0]

is (p(r)=r^2-7r+12=(r-3)(r-4)), so ({e^{3x},e^{4x}}) is a fundamental set of solutions of Equation ef{eq:5.4.3}. Therefore the general solution of Equation ef{eq:5.4.2} is

[y=2e^{2x}+c_1e^{3x}+c_2e^{4x}. onumber]

Example (PageIndex{2})

Find a particular solution of

[label{eq:5.4.4} y''-7y'+12y=5e^{4x}.]

Then find the general solution.

Solution

Fresh from our success in finding a particular solution of Equation ef{eq:5.4.2} — where we chose (y_p=Ae^{2x}) because the right side of Equation ef{eq:5.4.2} is a constant multiple of (e^{2x}) — it may seem reasonable to try (y_p=Ae^{4x}) as a particular solution of Equation ef{eq:5.4.4}. However, this will not work, since we saw in Example (PageIndex{1}) that (e^{4x}) is a solution of the complementary equation Equation ef{eq:5.4.3}, so substituting (y_p=Ae^{4x}) into the left side of Equation ef{eq:5.4.4}) produces zero on the left, no matter how we choose(A). To discover a suitable form for (y_p), we use the same approach that we used in Section 5.2 to find a second solution of

[ay''+by'+cy=0 onumber]

in the case where the characteristic equation has a repeated real root: we look for solutions of Equation ef{eq:5.4.4} in the form (y=ue^{4x}), where (u) is a function to be determined. Substituting

into Equation ef{eq:5.4.4} and canceling the common factor (e^{4x}) yields

[(u''+8u'+16u)-7(u'+4u)+12u=5, onumber]

or

[u''+u'=5. onumber]

By inspection we see that (u_p=5x) is a particular solution of this equation, so (y_p=5xe^{4x}) is a particular solution of Equation ef{eq:5.4.4}. Therefore

[y=5xe^{4x}+c_1e^{3x}+c_2e^{4x} onumber]

is the general solution.

Example (PageIndex{3})

Find a particular solution of

[label{eq:5.4.6} y''-8y'+16y=2e^{4x}.]

Solution

Since the characteristic polynomial of the complementary equation

[label{eq:5.4.7} y''-8y'+16y=0]

is (p(r)=r^2-8r+16=(r-4)^2), both (y_1=e^{4x}) and (y_2=xe^{4x}) are solutions of Equation ef{eq:5.4.7}. Therefore Equation ef{eq:5.4.6}) does not have a solution of the form (y_p=Ae^{4x}) or (y_p=Axe^{4x}). As in Example (PageIndex{2}), we look for solutions of Equation ef{eq:5.4.6} in the form (y=ue^{4x}), where (u) is a function to be determined. Substituting from Equation ef{eq:5.4.5} into Equation ef{eq:5.4.6} and canceling the common factor (e^{4x}) yields

[(u''+8u'+16u)-8(u'+4u)+16u=2, onumber]

or

[u''=2. onumber]

Integrating twice and taking the constants of integration to be zero shows that (u_p=x^2) is a particular solution of this equation, so (y_p=x^2e^{4x}) is a particular solution of Equation ef{eq:5.4.4}. Therefore

[y=e^{4x}(x^2+c_1+c_2x) onumber]

is the general solution.

The preceding examples illustrate the following facts concerning the form of a particular solution (y_p) of a constant coefficent equation

[ay''+by'+cy=ke^{alpha x}, onumber]

where (k) is a nonzero constant:

1. If (e^{alpha x}) isn’t a solution of the complementary equation [label{eq:5.4.8} ay''+by'+cy=0,] then (y_p=Ae^{alpha x}), where (A) is a constant. (See Example (PageIndex{1})).
2. If (e^{alpha x}) is a solution of Equation ef{eq:5.4.8} but (xe^{alpha x}) is not, then (y_p=Axe^{alpha x}), where (A) is a constant. (See Example (PageIndex{2}).)
3. If both (e^{alpha x}) and (xe^{alpha x}) are solutions of Equation ef{eq:5.4.8}, then (y_p=Ax^2e^{alpha x}), where (A) is a constant. (See Example (PageIndex{3}).)

See Exercise 5.4.30 for the proofs of these facts.

In all three cases you can just substitute the appropriate form for (y_p) and its derivatives directly into

[ay_p''+by_p'+cy_p=ke^{alpha x}, onumber]

and solve for the constant (A), as we did in Example (PageIndex{1}). (See Exercises 5.4.31-5.4.33.) However, if the equation is

[ay''+by'+cy=k e^{alpha x}G(x), onumber]

where (G) is a polynomial of degree greater than zero, we recommend that you use the substitution (y=ue^{alpha x}) as we did in Examples (PageIndex{2}) and (PageIndex{3}). The equation for (u) will turn out to be

[label{eq:5.4.9} au''+p'(alpha)u'+p(alpha)u=G(x),]

where (p(r)=ar^2+br+c) is the characteristic polynomial of the complementary equation and (p'(r)=2ar+b) (Exercise 5.4.30); however, you shouldn’t memorize this since it is easy to derive the equation for (u) in any particular case. Note, however, that if (e^{alpha x}) is a solution of the complementary equation then (p(alpha)=0), so Equation ef{eq:5.4.9} reduces to

[au''+p'(alpha)u'=G(x), onumber]

while if both (e^{alpha x}) and (xe^{alpha x}) are solutions of the complementary equation then (p(r)=a(r-alpha)^2) and (p'(r)=2a(r-alpha)), so (p(alpha)=p'(alpha)=0) and Equation ef{eq:5.4.9}) reduces to

[au''=G(x). onumber]

Example (PageIndex{4})

Find a particular solution of

[label{eq:5.4.10} y''-3y'+2y=e^{3x}(-1+2x+x^2).]

Solution

Substituting

into Equation ef{eq:5.4.10}) and canceling (e^{3x}) yields

[(u''+6u'+9u)-3(u'+3u)+2u=-1+2x+x^2, onumber]

or

[label{eq:5.4.11} u''+3u'+2u=-1+2x+x^2.]

As in Example 5.3.2, in order to guess a form for a particular solution of Equation ef{eq:5.4.11}), we note that substituting a second degree polynomial (u_p=A+Bx+Cx^2) for (u) in the left side of Equation ef{eq:5.4.11}) produces another second degree polynomial with coefficients that depend upon (A), (B), and (C); thus,

If (u_p) is to satisfy Equation ef{eq:5.4.11}), we must have

[egin{aligned} u_p''+3u_p'+2u_p&=2C+3(B+2Cx)+2(A+Bx+Cx^2) &=(2C+3B+2A)+(6C+2B)x+2Cx^2=-1+2x+x^2.end{aligned} onumber ]

Equating coefficients of like powers of (x) on the two sides of the last equality yields

[egin{array}{rcr} 2C&=1phantom{.} 2B+6C&=2phantom{.} 2A+3B+2C&= -1. end{array} onumber ]

Solving these equations for (C), (B), and (A) (in that order) yields (C=1/2,B=-1/2,A=-1/4). Therefore

[u_p=-{1over4}(1+2x-2x^2) onumber]

is a particular solution of Equation ef{eq:5.4.11}, and

[y_p=u_pe^{3x}=-{e^{3x}over4}(1+2x-2x^2) onumber]

is a particular solution of Equation ef{eq:5.4.10}.

Example (PageIndex{5})

Find a particular solution of

[label{eq:5.4.12} y''-4y'+3y=e^{3x}(6+8x+12x^2).]

Solution

Substituting

into Equation ef{eq:5.4.12}) and canceling (e^{3x}) yields

[(u''+6u'+9u)-4(u'+3u)+3u=6+8x+12x^2, onumber]

or

[label{eq:5.4.13} u''+2u'=6+8x+12x^2.]

There’s no (u) term in this equation, since (e^{3x}) is a solution of the complementary equation for Equation ef{eq:5.4.12}). (See Exercise 5.4.30.) Therefore Equation ef{eq:5.4.13}) does not have a particular solution of the form (u_p=A+Bx+Cx^2) that we used successfully in Example (PageIndex{4}), since with this choice of (u_p),

[u_p''+2u_p'=2C+(B+2Cx) onumber]

can’t contain the last term ((12x^2)) on the right side of Equation ef{eq:5.4.13}). Instead, let’s try (u_p=Ax+Bx^2+Cx^3) on the grounds that

together contain all the powers of (x) that appear on the right side of Equation ef{eq:5.4.13}).

Substituting these expressions in place of (u') and (u'') in Equation ef{eq:5.4.13}) yields

[(2B+6Cx)+2(A+2Bx+3Cx^2)=(2B+2A)+(6C+4B)x+6Cx^2=6+8x+12x^2. onumber]

Comparing coefficients of like powers of (x) on the two sides of the last equality shows that (u_p) satisfies Equation ef{eq:5.4.13}) if

[egin{array}{rcr} 6C&=12phantom{.} 4B+6C&=8phantom{.} 2A+2Bphantom{+6u_2}&=6. end{array} onumber ]

Solving these equations successively yields (C=2), (B=-1), and (A=4). Therefore

[u_p=x(4-x+2x^2) onumber]

is a particular solution of Equation ef{eq:5.4.13}), and

[y_p=u_pe^{3x}=xe^{3x}(4-x+2x^2) onumber]

is a particular solution of Equation ef{eq:5.4.12}).

Example (PageIndex{6})

Find a particular solution of

[label{eq:5.4.14} 4y''+4y'+y=e^{-x/2}(-8+48x+144x^2).]

Solution

Substituting

into Equation ef{eq:5.4.14}) and canceling (e^{-x/2}) yields

[4left(u''-u'+{uover4} ight)+4left(u'-{uover2} ight)+u=4u''=-8+48x+144x^2, onumber]

or

[label{eq:5.4.15} u''=-2+12x+36x^2,]

which does not contain (u) or (u') because (e^{-x/2}) and (xe^{-x/2}) are both solutions of the complementary equation. (See Exercise 5.4.30.) To obtain a particular solution of Equation ef{eq:5.4.15}) we integrate twice, taking the constants of integration to be zero; thus,

Therefore

[y_p=u_pe^{-x/2}=x^2e^{-x/2}(-1+2x+3x^2) onumber]

is a particular solution of Equation ef{eq:5.4.14}).

## Summary

​​​​​The preceding examples illustrate the following facts concerning particular solutions of a constant coefficent equation of the form

[ay''+by'+cy=e^{alpha x}G(x), onumber]

where (G) is a polynomial (see Exercise 5.4.30):

1. If (e^{alpha x}) isn’t a solution of the complementary equation [label{eq:5.4.16} ay''+by'+cy=0,] then (y_p=e^{alpha x}Q(x)), where (Q) is a polynomial of the same degree as (G). (See Example (PageIndex{4})).
2. If (e^{alpha x}) is a solution of Equation ef{eq:5.4.16} but (xe^{alpha x}) is not, then (y_p=xe^{alpha x}Q(x)), where (Q) is a polynomial of the same degree as (G). (See Example (PageIndex{5}).)
3. If both (e^{alpha x}) and (xe^{alpha x}) are solutions of Equation ef{eq:5.4.16}, then (y_p=x^2e^{alpha x}Q(x)), where (Q) is a polynomial of the same degree as (G). (See Example (PageIndex{6}).)

In all three cases, you can just substitute the appropriate form for (y_p) and its derivatives directly into

[ay_p''+by_p'+cy_p=e^{alpha x}G(x), onumber]

and solve for the coefficients of the polynomial (Q). However, if you try this you will see that the computations are more tedious than those that you encounter by making the substitution (y=ue^{alpha x}) and finding a particular solution of the resulting equation for (u). (See Exercises 5.4.34-5.4.36.) In Case (a) the equation for (u) will be of the form

[au''+p'(alpha)u'+p(alpha)u=G(x), onumber]

with a particular solution of the form (u_p=Q(x)), a polynomial of the same degree as (G), whose coefficients can be found by the method used in Example (PageIndex{4}). In Case (b) the equation for (u) will be of the form

[au''+p'(alpha)u'=G(x) onumber]

(no (u) term on the left), with a particular solution of the form (u_p=xQ(x)), where (Q) is a polynomial of the same degree as (G) whose coefficents can be found by the method used in Example (PageIndex{5}). In Case (c), the equation for (u) will be of the form

[au''=G(x) onumber]

with a particular solution of the form (u_p=x^2Q(x)) that can be obtained by integrating (G(x)/a) twice and taking the constants of integration to be zero, as in Example (PageIndex{6}).

## Using the Principle of Superposition

The next example shows how to combine the method of undetermined coefficients and Theorem 5.3.3, the principle of superposition.

Example (PageIndex{7})

Find a particular solution of

[label{eq:5.4.17} y''-7y'+12y=4e^{2x}+5e^{4x}.]

Solution

In Example (PageIndex{1}) we found that (y_{p_1}=2e^{2x}) is a particular solution of

[y''-7y'+12y=4e^{2x}, onumber]

and in Example (PageIndex{2}) we found that (y_{p_2}=5xe^{4x}) is a particular solution of

[y''-7y'+12y=5e^{4x}. onumber]

Therefore the principle of superposition implies that (y_p=2e^{2x}+5xe^{4x}) is a particular solution of Equation ef{eq:5.4.17}).

## 4.6: The Method of Undetermined Coefficients I - Mathematics

• Course Description: This course is intended to introduce students basic solution methods for ordinary differential equations. First order equations, linear equations, constant coefficient equations. Eigenvalue methods for systems of first order linear equations. Introduction to stability and phase plane analysis. Laplace transforms and series solutions to ordinary differential equations.

(3-0) Cr. 3. F.S.SS. Prereq : Minimum of C- in MATH 166 or MATH 166H

Solution methods for ordinary differential equations. First order equations, linear equations, constant coefficient equations. Eigenvalue methods for systems of first order linear equations. Introduction to stability and phase plane analysis.

• Textbook:Differential Equations and Boundary Value Problems Cengage, 9th edition, by Zill with access to WebAssign online homework platform
• Instructor:Hailiang Liu
• Office: Carver 434, phone: 294-0392, Email: [email protected]
• Office Hours: MWF 10 :00 A - 11 :30 A and also by appointment
• Lecture Meeting Time: MWF: 1:10P- 2:00P Carver 0202
• Homework:

We use WebAssign for the homework together with the text book . ISU book store and the textbook publisher have an immediate access'' program that lets you go to WebAssign using the link in Canvas at https://canvas.iastate.edu and here is how it works. You do not need a separate code to get in to WebAssign . Please click the link that says WebAssign in Canvas. It will take you to WebAssign website where your homework assignments are located. You will have to make an account in WebAssign (if you do not have one already). Once you do, you are ready to work on the available homework. Every time you want to work on the HWs, please follow the link in Canvas. Your HW scores will be synced that way.

Exam 1
on Friday Sept. 27.
Exam 2 on Friday Oct 25

Exam 3 on Friday November 22

Final is comprehensive and is on Finals week.

Calculators will not be allowed. Also you must show your work on the papers step by step to get full credit. No make-up exams, except for special circumstances. If an emergency causes you to miss an exam, contact the instructor as soon as possible.

Grading: Each midterm exam counts 15% of your grade, and homework and quiz count 30% and the final exam counts 25%. An appropriate scaling may be applied at the end of the semester to determine the final grade.

Course objectives for Math 266

o Be able to use the method of integrating factors to solve first order linear equations.

o Be able to separate variables and compute integrals in solving first order separable equations.

o Know how to find a general solution of a linear second order constant coefficient homogeneous differential equation by seeking exponential solutions.

o Be able to use the method of undetermined coefficients to find a particular solution of a linear second order constant coefficient nonhomogeneous differential equation.

o Be able to find a general solution of a linear second order constant coefficient nonhomogeneous equation.

o Be able to solve an initial value problem associated with a linear second order constant coefficient homogeneous or nonhomogeneous equation.

o Be able to extend the methods used for linear second order constant coefficient equations to higher order linear constant coefficient equations, both homogeneous and non-homogeneous.

o Be able to use the eigenvalue-eigenvector method to find general solutions of linear first order constant coefficient systems of differential equations of size 2 or 3.

o Be able to find a fundamental matrix for linear first order constant coefficient system of differential equations of size 2 or 3.

o Be able to use the method of variation of parameters to find a particular solution of a nonhomogeneous linear first order constant coefficient system of size 2.

Learn how differential equations are used to model physical systems and other applied problems. These could include the following types of problems .

o Be able to formulate and use elementary models for population dynamics, such as the logistic equation, to describe transient and steady state behavior.

o Be able to work with models for the linear motion of objects using assumptions on the velocity and acceleration of the object.

o Be able to set up and solve a problem involving stirred tank reactor dynamics.

o Be able to use Newton's second law to set up a model for a simple spring-mass system and use appropriate methods to obtain the solution of the model problem.

o Be able to use models for continuous compounding of interest to describe elementary savings and loan problems.

Gain an elementary understanding of the theory of ordinary differential equations.

o Understand statements on existence and uniqueness of solutions.

o Understand the role of linear independence of solutions in finding general solutions of differential equations.

o Understand what constitutes a general solution of a differential equation.

o Understand the concept of stability as it relates to equilibrium solutions.

Official Math Department Policies   The Math Department Class Policies page describes the official policies that all instructors have to follow. It covers rules on make-up exams, cheating, student behavior, etc .

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## Course outline

Instructor
Dr. Gantumur Tsogtgerel
Office: Burnside Hall 1123
Office hours: W 14:35󈝻:55, or by appointment
Email: gantumur -at- math.mcgill.ca

Note: The first three books are basically the same. The first book (Zill) is a subset of the second (Zill-Wright) and the third (Zill-Cullen), and we will not cover the additional chapters that are present in Zill-Wright and Zill-Cullen. The fourth book (Trench) is a free online book, and has a similar material as in Zill.

Webwork 15% + Written assignment 15% + max

Catalog description
First order ordinary differential equations including elementary numerical methods. Linear differential equations. Laplace transforms. Series solutions.

Prerequisite
MATH 222 (Calculus 3)

Corequisite
MATH 133 (Linear algebra and geometry)

Restriction
Not open to students who have taken or are taking MATH 325.

## 4.6: The Method of Undetermined Coefficients I - Mathematics

Course Name: Math 266, section 4, Fall 2014.

### Course Material

• Textbook Differential Equations with Boundary Value Problems, ISU Custom ed. by D.G. Zill and W. S. Wright
• Syllabus:
• Introduction and orientation (Chapter 1)
• First order equations (Chapters 2, 3)
• Second and higher order linear equations (Chapter 4,5)
• Systems of first order equations (Chapters 8,10)

### Course Policy

• Homework:
• The WebAssign online homework system will be used for homework submission and grading. The due date will be stated clearly with each set of problems. The online homework system does not accept late homework. I may also give you some more problems from the text from time to time that are not available in WebAssign. Most of these problems will be practice problems, not to be turned in.
• All the homeworks (assigned problems and practice problems) will be posted on the course log at the bottom of this page. You will be also able to see the assigned homeworks once you log into WebAssign.
• Quizzes and homeworks: 20% of the grade.
• First Midterm: 25% (Friday of Week 6)
• Second Midterm: 25% (Friday of week 12)
• Final: 30%.
• Cell phones and other communications devices should be turned off during class time. In the event that a student needs to keep his/her cell phone on for an emergency, please notify the instructor before class. During class time, laptop computers should only be used for class appropriate purposes.
• The final exam must occur at the time and place designated on the University Final Exam Schedule.
• There will be no makeup exams (midterm or final) except in rare and extreme circumstances (based on the instructor's discretion). In these cases written proof of valid reason of absence will be mandatory.

### Course objevtives

• This is to keep track of what has to be covered, what has been covered so far, to remedy the things that I might blotch up in class. This is also where I shall put up the homeworks.

Section 1.2: # 1, 3, 7, 9, 11 (due in WebAssign on Friday Sept 5, 11:59 pm).

MIT open courseware lecture 1 (direction fields etc), Here is the link to the MIT opencourseware ODE course main page where lot of material can be found.

Section 2.2, # 2, 6, 9, 16, 22, 25
Section 2.3, # 7, 10, 16, 27, 34
Section 2.4, # 2, 5, 8, 9, 14, 23
Section 2.5, # 3, 5, 12, 13, 16, 24, 25

Section 4.1, # 7, 13, 19, 32.
Section 4.3, # 2, 4, 12, 16, 21, 23, 30, 34
Section 4.4, # 4, 5, 8, 12, 15, 20, 21, 30
Section 4.6, # 5, 10, 15, 20
Section 4.7, # 3, 9, 12, 21

Section 8.1, # 2, 9, 25.
Section 8.2 # 2, 10, 19, 27, 41.
Section 8.3 # 2, 6, 17, 26.

## 15.3. Method 2: use monomials of degree up to (p+k-1) ¶

From the above degree of precision result, one can determine the coefficients by requiring degree of precision (p+k-1) , and for this it is enough to require exactness for each of the simple monomial functions (1) , (x) , (x^2) , and so on up to (x^) .

Also, this only needs to be tested at (x=0) , since “translating” the variables does not effect the result.

This is probably the simplest method in practice.

Example 4 (Example 2 revisited)

The goal is to get exactness in

for the monomials (f(x) = 1) , (f(x) = x) , and so on, to the highest power possible, and this only needs to be checked at (x=0) .

We need at least three equations for the three unknown coefficients, so continue with (f(x) = x^2) , (Df(0) = 0) :

We can solve these by elimination for example:

The last equation gives (C_1 = -4C_2)

The previous one then gives (-4C_2 + 2C_2 = 1) , so (C_2 = -1/2) and thus (C_1 = -4C_2 = 2) .

The first equation then gives (C_0 = -C_1 - C_2 = -3/2) all as claimed above.

So far the degree of precision has been shown to be at least 2. In some cases it is better, so let us check by looking at (f(x) = x^3) :

So, no luck this time (that typically requires some symmetry), but this calculation does indicate in a relatively simple way that the error is (O(h^2)) .

If you want to verify more rigorously the order of accuracy of a formula devised by this method, one can use the “checking” procedure with Taylor polynomials and their error terms as done in Example 2 above.

### 15.3.1. Exercise 2: like Exercise 1, but using Method 2¶

#### 15.3.1.1. A)¶

Verify the result in Example 2, this time by Method 2.

That is, impose the condition of giving the exact value for the derivative at (x=0) for the monomial (f(x) = 1) , then the same for (f(x) = x) , and so on until there are enough equations to determine a unique solution for the coefficients.

 MA 101 : Mathematics I (4-1-0-4) Review of limits, continuity, differentiability Mean value theorem, Taylor’s Theorem, Maxima and Minima Riemann integrals, Fundamental theorem of Calculus, Improper integrals, applications to area, volume Convergence of sequences and series, Newton’s method, Picard’s method Multi-variable functions, Partial Derivatives, gradient and directional derivatives, chain rule, maxima and minima, Lagrange multipliers Double and Triple integration, Jacobians and change of variables formula Parametrization of curves and surfaces, vector fields, Line and surface integrals Divergence and curl, Theorems of Green, Gauss, and Stokes. MA 102 : Mathematics II (3-1-0-4) Linear Algebra: Vectors in Rn Vector subspaces of Rn Basis of vector subspace Systems of Linear equations Matrices and Gauss elimination Determinants and rank of a matrix Abstract vector spaces, Linear transformations, Matrix of a linear transformation, Change of basis and similarity, Rank-nullity theorem Inner product spaces, Gram-Schmidt process, Orthonormal bases Projections and least-squares approximation Eigenvalues and eigenvectors, Characteristic polynomials, Eigenvalues of special matrices Multiplicity, Diagonalization, Spectral theorem, Quadratic forms. Differential Equations: Exact equations, Integrating factors and Bernoulli’s equation Orthogonal trajectories Lipschitz condition, Picard’s theorem Wronskians Dimensionality of space of solutions, Abel-Liouville formula Linear ODE’s with constant coefficients Cauchy-Euler equations Method of undetermined coefficients Method of variation of parameters Laplace transforms, Shifting theorems, Convolution theorem. MA 201 : Mathematics III (3-1-0-4) Complex Analysis: Definition and properties of analytics functions Cauchy-Riemann equations, Harmonic functions Power series and their properties Elementary functions Cauchy’s theorem and its applications Taylor series and Laurent expansions Residues and the Cauchy residue formula Evaluation of improper integrals Conformal mappings. Differential Equations: Review of power series and series solutions of ODE’s Legendre’s equation and Legendre polynomials Regular and irregular singular points, method of Frobenius Bessel’s equation and Bessel’s functions Sturm-Liouville problems Fourier series D’Alembert solution to the Wave equation Classification of linear second order PDE in two variables Vibration of a circular membrane Fourier Integrals, Heat equation in the half space. MA 202 : Mathematics IV (3-2-0-4) Probability and Statistics

Random Experiments Events Probability Random variables Probability Distributions: Discrete and Continuous Distributions, Mean and Variance of Distributions, Distributions of Several Random variables Random sampling Estimation of parameters Confidence Intervals Testing of Hypotheses Goodness of fit – test Quality control and Acceptance Sampling Confidence intervals for regression parameters.

Separation Axioms: Hausdorff spaces, Regularity, Complete Regularity, Normality, Urysohn Lemma, Tychonoff embedding and UrysohnMetrization Theorem,

Tietze Extension Theorem, Tychnoff Theorem, One-point Compactification.

Complete metric spaces and function spaces, Characterization of compact metric spaces, equicontinuity, Ascoli-Arzela Theorem, Baire Category Theorem.

Applications: space filling curve, nowhere differentiable continuous function,

Interpolation Formulas of Lagrange and Newton, Error in Polynomial Interpolation, Hermite Interpolation, Interpolation by Spline Functions, Cubic Spline and B-Splines.

Module 2, System of Linear Equations:

Gaussian Elimination, Gauss-Jordan Algorithm, The Cholesky Decomposition, Q-R Decomposition, Least Square Approximations, Iterative Methods and Convergence Theorems.

Module 3, Nonlinear Equations in R^n:

Derivatives and Other Basic Concepts, Convex Functionals, Contractions, Inverse and Implicit Function Theorems, Newton’s Methods and Its Variations and Minimization Methods.

Module 4, Differential Equations and Boundary Value Problems:

One Step Methods with Convergence, Multi Step Methods with Convergence, Simple and Multiple Shooting Methods, Difference Method and Variational Method.

## Further Information

• Honor Code: The Honor Code applies to all work for this course. Please review the Honor Code at [this link]. Students found violating the Honor Code will be subject to discipline.
• Some material will be stored in Dropbox. In that case, you will need an account to retrieve it. If you do not have one already, sign-in through [this link] with your academic e-mail address to receive a base 4GB storage, plus an extra 500MB, free of charge.
• Remember to change your e-mail address on Blackboard if necessary [blackboard.sc.edu]
• ADA: If you have special needs as addressed by the Americans with Dissabilities Act and need any assistance, please notify the instructor immediately.
• Peer Tutoring: Tutoring is available for this course to assist you in better understanding the course material. The Peer Tutoring Program at the Student Success Center provides free peer-facilitated study sessions led by qualified and trained undergraduate tutors who have previously taken and excelled in this course. Sessions are open to all students who want to improve their understanding of the material, as well as their grades. Tutoring is offered Sunday 6-10pm and Monday through Thursday 2-9pm. All tutoring sessions will take place on the Mezzanine Level of the Thomas Copper Library unless otherwise noted. Please visit www.sc.edu/tutoring to find the complete tutoring schedule and make an appointment. You may also contact the Student Success Center at 803-777-1000 and [email protected] with additional questions. The tutor for your course is Alexandra Ruppe

## 4.6: The Method of Undetermined Coefficients I - Mathematics

Instructor: Dr. Jessica M. Conway.
Lectures: MWF 1-2pm, Henn 200.
Office hours: Mathematics Annex, Room 1110 - Wednesdays 4-6pm + by appointment.
OFFICE HOURS DURING FINAL EXAMS (DEC 6-20): Tuesdays/Thursdays from 3-5pm Saturday Dec 18/Sunday Dec 19 from 4-7pm.
Email: conway (at) math (dot) ubc (dot) ca OR math255s104.fall2010 (at) gmail (dot) com
Phone: (604)822-6754
The Mathematics Department offers Drop in tutoring, ODEs included!

Text: Boyce and Diprima, Elementary differential equations and boundary value problems, 9th edition.
We will cover chapters 1-3, 6, 7, and 9.
Note: If you have instead an 8th edition of the text, that's fine. Problems and readings for the 8th edition are also provided below.

### Homework

Homework 2 , due Sept 24th 2010:
9th edition: p.47: 34 p.59: 32 p.75: 3 p.88: 15,22 p.99: 13.
OR 8th edition: p.47: 34 p.59: 32 p.75: 3 p.88: 15,22 p.99: 13.
SOLUTIONS here

Homework 3 , due Oct 1st 2010:
9th edition: p.144: 1, 9, 13, 23 p.155: 1.
OR 8th edition: p.142: 1, 9, 13, 23 p.151: 1.
SOLUTIONS here

Homework 4 , due Oct 8th 2010:
9th edition: p.163: 2, 17, 29, 32 p.171: 23 p.183: 17, 28.
OR 8th edition:p.164: 2, 17, 29, 32 p.173: 23 p.184: 17, 28.
SOLUTIONS here

Homework 5 , due WEDNESDAY Oct 20th 2010:
9th edition: p.189: 1, 19, 21, 28 p.202: 5, 15, 16 p.216: 17.
OR 8th edition:p.190: 1, 19, 21, 28 p.203: 5, 15, 16 p.214: 17.
SOLUTIONS here

Homework 6 , due Friday October 29th 2010:
9th edition: p.311: 14, 18, 26 p.320: 27a p.328: 13, 25, 29, 30.
OR 8th edition:p.312: 14, 18, 26 p.322: 27a p.329: 7, 19, 23, 24.
SOLUTIONS here

Homework 7 , due Friday November 5th 2010:
9th edition: p.336: 1, 10 p.343: 25 p.350: 7, 13, 22, 29.
OR 8th edition: p.337: 1, 10 p.344: 25 p.351: 7, 13, 22, 29.
Note: p.351: 22b,c and 29 will not be graded.
SOLUTIONS here and, for 6.4.1 and 6.4.10, here.

Homework 8 , due Friday November 12th 2010:
9th edition:p.398: 15, 28, 29, 32, 33 p.409: 26, 27 p.428: 1.
OR 8th edition: p.398: 15, 28, 29, 32, 33 p.410 26, 27 p.428: 1.
SOLUTIONS here and, for 7.8.1, here.

Homework 9 , due Friday November 19th 2010:
9th edition: p.439: 1,3 p494: 2(a)-(c).
OR 8th edition: p.439: 1,3 p492: 2(a)-(c).
SOLUTIONS here and, for 7.9.3: undetermined vectors, variation of vectors.

Homework 10 , due Friday November 26th 2010:
9th edition: p.494: 3, 4, 5 Page 506: 19.
OR 8th edition: p.492: 3, 4, 5 Page 501: 17..
SOLUTIONS here.

Homework 11 , due Wednesday Dec 1st 2010:
9th edition: p.516: 5,6, 19, 27, 30.
OR 8th edition: p.511: 5,6, 19, 26, 28.
SOLUTIONS here.

## 4.6: The Method of Undetermined Coefficients I - Mathematics

The book is Elementary Differential Equations and Boundary Value Problems The authors are Boyce and DiPrima

Please note This is the tenth edition! (It seems to differ little from earlier editions, so you are probably safe with them. I have the 9th as well so I can help you align the exercises)

### Software

Some of the exercises will require you to use a computer to create pictures. There are several ways to do this. Some of you are probably familiar with MatLab which has something called PPLANE which will be helpful. I wrote a software package XPPAUT for solving and graphing differential equations. This runs on all PCs and also runs on iOS devices (sorry, no Android). You can get this at This site . I can help you get it on your computer as it requires a small amount of effort

### Syllabus:

Homework due: 9/5: 1.1:15-20,23,24,26,29,302.1:13,15,16,31,322.2:1,5,8,9,10,17,31,37,36

• Here is a handout for using XPP and doing some of the computer problems How to plot
• Simple XPP code for direction fields
• Run this in XPP. Click on (D)ir.field (S)caled and then Return to accept the default. See the nice direction fields!
• Click (I)nitialconds m(I)ce and click around on the screen near the dashed line. See the trajectories. Tap ESC when done.
• Take a screen shot of this to print it if you want
Homework Due 9/26
• 3.1:1,7,9,12,17,20,23,28
• 3.2:1,2,4,7,12,13,16,17,23,28,29
• 3.3:1,4,6,10,15,21,34,35,39
• Homework Due 10/3
• 3.4:7,11,12,17,20,21
• 3.5: 1,6(4),10(8),14(12),16(14),20(18) [Note 9th edition in parentheses]
• 3.6: 1,2,9,13
• 3.7: 1,5,7,13,18
• 3.8: 1,11,17,24
• 4.1:3,6,7,11,15,24 (you can assume without proving it, the result of problem 20 on page 225)
• Read pages 3-5 of this handout
• Sample exam 1 (note problem 4 is page 157 in the 10th edition)
• Review 10/6
• Phase line (2.5)
• Direction fields (1.1)
• Applications of 1D linear (2.3)
• Methods of solving:
• Linear 1st order (2.1)
• Bernoulli equations (2.4 exercise 27)
• Exact equations (2.6)
• homogeneous equations (2.2 exercise 29)
• Second order linear equations (3.1-3.3)
• Chapter 4. 4.2:11,18,214.3:1(see 4.2,11)4.4:1
• Chapt 7.1 1,4,6,23
• 7.3:1,4,15,18,21,23
• 7.4:2abc,6
• 7.5:1,2,5,7,11,15,16,20,24,25,27,31
• 7.6:1,3,5,13,14,28
• 7.7:1,3,5
• 7.8:1,2,11
• Let A be a 3x3 matrix with eigenvalues -1, -1+2 i. Express exp(At) in terms of the matrix A. (Use Fulmer's method)
• Use Fulmer's method to do problem 1 in 7.7
• 7.9:3,12
• 9.1: 1,3,5,6,13
• 9.2:1,4,5,9,17,21

They have infinitely many periodic solutions. Let T be the period of one of the solutions and let x(t),y(t) be the solution. Compute the average values of x(t),y(t):

## MATH 351 (Spring 2014): Differential Equations

Midterm Exam 1: (date to be announed) in class.
Midterm Exam 2: (date to be annouced) in class.
Midterm Exam 3: (date to be annouced) in class.

### Lecture times and locations

Mondays & Wednesdays 11:00 am - 12:15 pm in LO (Live Oak Hall) 1326

### Course text

Elementary Differential Equations and Boundary Value Problems (10th edition), or the short version of Elementary Differential Equations by William E. Boyce and Richard C. DiPrima.

### Course syllabus and tentative timetable

I will post all assigments, solutions and additional material in this space. You should therefore consult this spot frequently.

Sec. 1.3: Classification of Differential Equations

Exercises 1.3 (page 24): 1, 3, 5, 7, 11, 14, 17, 19

(due Feb 5) Exercises 1.2 (page 15): 4, 5, 6, 8, 10, 12, 14, 16, 19

Exercises 1.3 (page 24): 2, 4, 6, 8, 10, 12, 13, 16, 18, 20, 29

Exercises 2.2 (page 48): 2-26 (even), 23, 32(a)(b), 34(a)(b), 36(a)(b), 38(a)(b)

Sec. 2.3: Modeling with First Order Equations

Exercises 2.3 (page 60): 1, 4, 8, 16, 19, 24

Exercises 2.4 (page 76): 1, 3, 7, 9, 11, 15, 22, 28, 33

Exercises 2.4 (page 76): 2, 4, 5, 6, 8, 10, 12, 13, 21, 23-26, 29

(Exercises 2.6 due Mar 5) Exercises 2.6 (page 101): 2-12 (even), 16, 20, 26, 28, 30

Sec. 2.7: Numerical Approximations: Euler's Method

Sec. 2.8: The Existence and Uniqueness Theorem

Sec. 3.1: Homogeneous Equations with Constant Coefficient

Sec. 3.2: Solutions of Linear Homogeneous Equations the Wronskian

Exercises 3.1 (page 144): 10, 11, 12, 15, 17, 19, 21, 23, 25, 27

Exercises 3.2 (page 155): 5, 6, 9, 11, 12, 14, 15, 17, 19, 22, 26, 31, 34

(Exercises 3.1 due March 24)

Miscellaneous Problems for Chapter 2 (page 133): 1-35

Exercises 3.1 (page 144): 1-7, 9, 13, 14, 16, 18, 20, 22, 24, 26, 28

Exercises 3.2 (page 155): 2, 3, 4, 7, 8, 10, 13, 16, 18, 20, 21, 23, 24, 25, 27, 28, 29, 30, 32, 33, 35

Sec. 3.4: Repeated Roots Reduction of Order

Exercises 3.3 (page 164): 5, 8, 9, 12, 13, 18, 19, 20, 25, 34, 39, 40

Exercises 3.6 (page 190): 3, 5, 7, 8, 9, 11, 13, 15, 17, 31 5.1 (page 253): 2, 3, 5, 6, 7, 8, 12, 13, 15, 17, 21, 22, 23, 24, 27 -->

Sec. 4.1: General Theory of nth Order Linear Equations

Sec. 4.2: Homogeneous Equations with Constant Coefficients

Sec. 4.3: The Method of Undetermined Coefficients

Exercises 4.2 (page 233): 2, 5, 9, 11, 13, 15, 16, 19, 20 Exercises 4.3 (page 239): 1, 4, 5, 6, 9, 11, 13, 15, 17

Exercises 4.4 (page 244): 1, 2, 3, 9 5.2 (page 263): 4, 7, 8, 9, 10, 11, 12, 19, 21

Exercises 5.3 (page 269): 2, 3, 6, 7, 10, 12, 13, 16, 19, 22, 23

(Exercises 4.2 due April 16 )

(Exercises 4.3 due April 16)

Exercises 4.2 (page 233): 1, 3, 4, 6, 10, 12, 14, 17, 18, 21-24

Exercises 4.3 (page 239): 3, 8, 10, 14, 16, 18

Exercises 4.4 (page 244): 4, 5, 7, 11, 13 5.2 (page 263): 5, 6, 13, 14, 20, 22

Exercises 5.3 (page 269): 1, 4, 5, 8, 9, 11, 14, 17, 18, 20, 21

Sec. 7.2: Review of Matrices

Sec. 7.3: Systems of Linear Algebraic Equations: Linear Independence, Eigenvalues, Eigenvectors

Exercises 7.2 (page 376): 2, 8, 9, 11, 13, 15, 17, 21, 23, 25

Exercises 7.3 (page 388): 2, 3, 4, 5, 7, 11, 13, 16, 17, 22, 23, 25

(Exercises 7.2 due April 23)

Exercises 7.2 (page 376): 1, 3, 4, 5, 6, 7, 10, 12, 14, 16, 18, 19, 20, 22, 24, 26

Exercises 7.3 (page 388): 6, 8, 9, 10, 14, 18, 19, 20, 21, 24

Sec. 7.5: Homogeneous Linear Systems with Constant Coefficients

Sec. 7.6: Complex Eigenvalues

Sec. 7.7: Fundamental Matrices

Exercises 7.5 (page 405): 2, 3, 6, 7, 9, 10, 11, 13, 16, 17, 25, 27 7.6 (page ): -->

Exercises 7.6 (page 417): 4, 5, 6, 7, 9, 19

Exercises 7.7 (page 427): 1, 3, 5, 7, 9, 11

(Exercises 7.5 due April 30)

Exercises 7.4 (page 394): 1, 3, 7

Exercises 7.5 (page 405): 1, 4, 5, 8, 12, 14, 15, 18-24, 26, 28, 29, 30

7.6 (page ): --> Exercises 7.6 (page 417): 1, 2, 3, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20

Exercises 7.7 (page 427): 2, 4, 6, 8, 10, 12, 17

Exercises 7.8 (page 436): 1, 6, 8, 10, 11, 12, 15

Sec. 9.2: Autonomous Systems and Stability

Sec. 9.3: Locally Linear Systems

Exercises 9.2 (page 517): 1, 2, 3, 4, 17, 19, 21

Exercises 9.3 (page 527): 1, 3, 5, 6, 7, 12, 15, 16, 19, 21, 26, 27, 28

Exercises 9.4 (page 541): 1, 3, 5, 8, 9, 10

Exercises 9.2 (page 517): 5-16, 18, 20, 22, 23, 24, 25, 26, 27, 28

Exercises 9.3 (page 527): 2, 4, 8, 9, 10, 11, 13, 14, 17, 18, 20, 23, 24, 25, 30