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

[label{eq:5.4.5} y=ue^{4x},quad y'=u'e^{4x}+4ue^{4x},quad ext{and} quad y''=u''e^{4x}+8u'e^{4x}+16ue^{4x}]

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:

- 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})).
- 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}).)
- 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

[y=ue^{3x},quad y'=u'e^{3x}+3ue^{3x},quad ext{and} y''=u''e^{3x}+6u'e^{3x}+9ue^{3x} onumber ]

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,

[ ext{if} quad u_p=A+Bx+Cx^2quad ext{then} quad u_p'=B+2Cxquad ext{and} quad u_p''=2C. onumber]

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

[y=ue^{3x},quad y'=u'e^{3x}+3ue^{3x},quad ext{and } y''=u''e^{3x}+6u'e^{3x}+9ue^{3x} onumber]

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

[u_p'=A+2Bx+3Cx^2quad ext{and} quad u_p''=2B+6Cx onumber ]

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

[y=ue^{-x/2},quad y'=u'e^{-x/2}-{1over2}ue^{-x/2},quad ext{and} quad y''=u''e^{-x/2}-u'e^{-x/2}+{1over4}ue^{-x/2} onumber]

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,

[u_p'=-2x+6x^2+12x^3quad ext{and} quad u_p=-x^2+2x^3+3x^4=x^2(-1+2x+3x^2). onumber]

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*):

- 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})).
- 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}).)
- 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

**Fall** **[email protected]**

**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]
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**Homework:****Additional Information and Resources****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)
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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.

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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.

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## 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!

Schedule and locations available at http://www.math.ubc.ca/Ugrad/ugradTutorials.shtml.**Printable course outline available HERE**.**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.### ANNOUNCEMENTS:

### Exam Dates:

### Grading

### 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

- 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.
- 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.### Announcements

### 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

## Watch the video: Tidy Up Your Home: The KonMari Method: Greeting and introduction (December 2021).

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 .

**Accessibility Statement**
Iowa State University is committed to assuring that all educational activities are free from discrimination and harassment based on disability status. Students requesting accommodations for a documented disability are required to work directly with staff in Student Accessibility Services (SAS) to establish eligibility and learn about related processes before accommodations will be identified.
After eligibility is established, SAS staff will create and issue a Notification Letter for each course listing approved reasonable accommodations. This document will be made available to the student and instructor either electronically or in hard-copy every semester. Students and instructors are encouraged to review contents of the Notification Letters as early in the semester as possible to
identify a specific, timely plan to deliver/receive the indicated accommodations. Reasonable accommodations are not retroactive in nature and are not intended to be an unfair advantage.

## 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.

Grading

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.