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Boyce - Elem. Diff. Eqs. & Boundary Value Problems (Homework)

James Finch

Math - College, section 1, Fall 2019

Instructor: Dr. Friendly

Current Score : 0 / 45

Due : Sunday, January 27, 2030 00:00 EST

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  • Instructions

    Elementary Differential Equations and Boundary Value Problems 10/e by William Boyce and Richard DiPrima published by John Wiley & Sons, Inc. is now available in WebAssign. The WebAssign component includes a complete online version of the text and a variety of interactive study aids. All of these rich resources are linked together in a dynamic and interactive environment for students. Additional links to the eBook within each problem are structured to support the way students work and provide them with superior just-in-time learning resources, and is available at a lower cost than the printed version.

    Question 2 has a GO tutorial that walks the student through how to find the particular solution of the given first order differential equation.

    Question 3 walks the student through how to show all solutions of a given differential equation approach a limiting value.

    Question 5 demonstrates interval grading, which can grade any canonically equivalent interval and enforce proper notation.

    Question 6 uses the matrix tool, which allows student to define the size of the resulting matrix.

    Question 7 showcases grading for bases, which accepts any correct eigenvector. Try multiplying by a constant!

    Question 8 showcases the grading of a differential equation involving vectors.

    Question 9 is an example of an in-depth look at how series questions are handled in WebAssign.

    Questions 10 and 11 show some of WebAssign's more intricate capabilities for handling the complex question types asked in Differential Equations.

    View the complete list of WebAssign questions available for this textbook. This demo assignment allows many submissions and allows you to try another version of the same question for practice wherever the problem has randomized values.

    The answer key and solutions will display after the first submission for demonstration purposes. Instructors can configure these to display after the due date or after a specified number of submissions.

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1. 0/4 points  |  Previous Answers BoyceDiffEQ10 2.1.001. My Notes
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Consider the following differential equation. (A computer algebra system is recommended.)
y' + 3y = t + e−2t
(a) Draw a direction field for the given differential equation.

Incorrect: Your answer is incorrect.

(b) Based on an inspection of the direction field, describe how solutions behave for large t.
     Incorrect: Your answer is incorrect.

(c) Find the general solution of the given differential equation.
y(t) =
0
Incorrect: Your answer is incorrect. webMathematica generated answer key


Use it to determine how solutions behave as
t → ∞.
     Incorrect: Your answer is incorrect.


Solution or Explanation
(a)
WebAssign Plot

(b) All solutions seem to approach a line in the region where the negative and positive slopes meet each other.

(c)    
μ = e∫3 dt = e3t.
 Thus,
e3t(y' + 3y)= e3t(t + e−2t)
 or
(ye3t)' = te3t + et.
 Integration of both sides yields
ye3t = 
t
3
e3t − 
1
9
e3t + et + C,
where integration by parts is used on the right side, with
u = t
 and
dv = e3t.
 Division by
e3t
 gives
y(t) = Ce−3t + 
t
3
 − 
1
9
 + e−2t,
 so y approaches
y = 
t
3
 − 
1
9
 as
t → ∞.
 This is the line identified in part (b).

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2. –/1 points BoyceDiffEQ10 2.1.016.GO. My Notes
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–/1
 
Find the solution of the given initial value problem.
y' + 
2
t
y = 
(cos t)
t2
,    y(π) = 0,  t > 0
y(t) =
sin()


GO Tutorial

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3. –/7 points BoyceDiffEQ10 2.1.032. My Notes
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–/7
 
Show that all solutions of
8y' + ty = 8
 approach a limit as
t → ∞.
 Hint: Consider the general solution,
y = e−t2/16
t
0
es2/16 ds + Ce−t2/16,
 and use L'Hôpital's rule on the first term.
Write the first term of
y = e−t2/16
t
0
es2/16 ds + Ce−t2/16
 as
t
0
 es2/16 ds
.
Use L'Hôpital's rule on the first term:
lim t → ∞ 
t
0
 es2/16 ds
 = lim t → ∞ 
et2/16
 = lim t → ∞ 
.

As
t → ∞
 the second term approaches
lim t → ∞ Ce−t2/16 =
.
 As t becomes large, all solutions converge to the function
.

Find the limiting value.

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4. 0/2 points  |  Previous Answers BoyceDiffEQ10 2.2.023. My Notes
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0/2
 
The result requested can be obtained either by solving the given equations analytically or by plotting numerically generated approximations to the solutions. Try to form an opinion about the advantages and disadvantages of each approach. (A computer algebra system is recommended.)

Solve the initial value problem.
y' = 9y2 + xy2,    y(0) = 1
y(x) =
2
Incorrect: Your answer is incorrect. webMathematica generated answer key


Determine where the solution attains its minimum value.
x =
2
Incorrect: Your answer is incorrect. webMathematica generated answer key


Solution or Explanation
Write the differential equation as
1
y2
 dy = (9 + x) dx.
 Integrating both sides, we obtain
−y−1 = 9x + 
1
2
x2 + C.
 Based on the specified initial condition, the solution can be written as
−y−1 = 9x + 
1
2
x2 − 1.
 Rearrange the equation to get
y = 
−1
x2
2
 + 9x − 1
.
 To determine where the solution attains its minimum value, find the roots of the equation
y' = 0.
 Referring back to the differential equation,
y' = 0
 for
x = −9.
 It can be seen that
y'' = (9y2 + xy2)' = 18yy' + y2 + 2xyy'.
 Therefore, recalling that
y'(−9) = 0,
 we conclude that
y''(−9) = (y(−9))2
 = 
2
92 + 2
2
 
≥0.
Hence the solution attains a minimum at
x = −9.

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5. –/1 points BoyceDiffEQ10 2.4.004. My Notes
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–/1
 
Determine (without solving the problem) an interval in which the solution of the given initial value problem is certain to exist. (Enter your answer using interval notation.)
(25 − t2)y' + 5ty = 8t2,   y(−8) = 1

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6. –/3 points BoyceDiffEQ10 7.2.004. My Notes
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–/3
 
If
A = 
9 − 7i  1 + i
7 − i  −7 + 9i
,
 find the following.
(a)    
AT



(b)    
A



(c)    
A*

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7. –/4 points BoyceDiffEQ10 7.3.016. My Notes
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Find all eigenvalues and eigenvectors of the given matrix. (Order eigenvalues from smallest to largest real part, then by imaginary part.)
7  −1
5  1
λ1 =
     		has eigenvector   
λ2 =
     		has eigenvector   

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8. 0/1 points  |  Previous Answers BoyceDiffEQ10 7.5.011. My Notes
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0/1
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0/1
 
Find the general solution of the given system of equations.
x' = 
1  1  7
1  7  1
7  1  1
x
x(t) =
s
Incorrect: Your answer is incorrect. webMathematica generated answer key


Solution or Explanation
We assume that
x = ξert
 and substitute for x in the given system. The result is the system of algebraic equations
1 − r  1  7
1  7 − r  1
7  1  1 − r
ξ1
ξ2
ξ3
 = 
0
0
0
.
This system has a nontrivial solution if and only if the determinant of the coefficient matrix is zero. Thus allowable values of r are found from the equation
1 − r  1  7
1  7 − r  1
7  1  1 − r
 = −r3 + 9r2 + 36r − 324 = (−r + 9)(r − 6)(r + 6) = 0.
The roots of this equation are
r1 = 9,
r2 = 6,
r3 = −6.
 Setting
r = 9,
 we have
−8  1  7
1  −2  1
7  1  −8
ξ1
ξ2
ξ3
 = 
0
0
0
.
This system is reduced to the equations
ξ1 − ξ2 = 0
ξ1 − ξ3 = 0.
So the corresponding eigenvector is
ξ(1) = 
1
1
1
.
Setting
r = 6,
 the reduced system of equations is
2ξ1 + ξ2 = 0
ξ1 − ξ3 = 0,
and the corresponding eigenvector is
ξ(2) = 
1
−2
1
.
Similarly, for
r = −6
 we have the reduced system
ξ1 + ξ3 = 0
ξ2 = 0,
and the corresponding eigenvector is
ξ(3) = 
1
0
−1
.
Since the eigenvalues are distinct, the general solution is
x = C1
1
1
1
e9t + C2
1
−2
1
e6t + C3
1
0
−1
e−6t,
where
C1,
C2,
 and
C3
 are arbitrary constants.

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9. –/7 points BoyceDiffEQ10 5.5.003. My Notes
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–/7
 
Consider the following differential equation.
xy'' + y = 0
(a) Show that the given differential equation has a regular singular point at
x = 0.
Note that
xp(x) =
and
x2q(x) =
,
 which are both analytic at
x = 0.
(b) Determine the indicial equation. (Enter your answer in terms of r.)


Determine the recurrence relation.
an =


Determine the roots of the indicial equation. (Enter your answers as a comma-separated list.)
r =


(c) Find the series solution
(x > 0)
 corresponding to the larger root.
y1(x) = 
∞
n = 0


(d) If the roots are unequal and do not differ by an integer, find the series solution corresponding to the smaller root also. (If the roots are equal or differ by an integer, enter DNE.)
y2(x) = 
∞
n = 0

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10. 0/6 points  |  Previous Answers BoyceDiffEQ10 9.1.001. My Notes
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0/6
 
Consider the following system. (A computer algebra system is recommended.)
dx
dt
 = 
3  −4
1  −2
x
(a) Find the eigenvalues and eigenvectors. (Order eigenvalues from smallest to largest real part, then by imaginary part. For repeated eigenvalues, list the eigenvalue once and give a maximal set of linearly independent eigenvectors.)
r1 =
2
Incorrect: Your answer is incorrect. webMathematica generated answer key        		has eigenvector    ξ(1) =
Incorrect: Your answer is incorrect. seenKey

[1; 1]
r2 =
−1
Incorrect: Your answer is incorrect. webMathematica generated answer key        		has eigenvector    ξ(2) =
Incorrect: Your answer is incorrect. seenKey

[4; 1]

(b) Classify the critical point
(0, 0)
 as to type, and determine whether it is stable, asymptotically stable, or unstable.
    

(c) Sketch several trajectories in the phase plane, and also sketch some typical graphs of
x1
 versus t. Use a computer to plot accurately the curves requested.

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11. –/9 points BoyceDiffEQ10 9.4.013. My Notes
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–/9
 
Consider the system
x' = F(x, y, α),    y' = G(x, y, α),
where α is a parameter. The equations
F(x, y, α) = 0,    G(x, y, α) = 0
determine the x- and y-nullclines, respectively; any point where an x-nullcline and a y-nullcline intersect is a critical point. As α varies and the configuration of the nullclines changes, it may well happen that, at a certain value of α, two critical points coalesce into one. For further variation in α, the critical point may once again separate into two critical points, or it may disappear altogether. Or the process may occur in reverse: For a certain value of α, two formerly nonintersecting nullclines may come together, creating a critical point, which, for further changes in α, may split into two. A value of α at which such phenomena occur is a bifurcation point. It is also common for a critical point to experience a change in its type and stability properties at a bifurcation point. Thus both the number and the kind of critical points may change abruptly as α passes through a bifurcation point. Since a phase portrait of a system is very dependent on the location and nature of the critical points, an understanding of bifurcations is essential to an understanding of the global behavior of the system's solutions.

Consider the following system. (A computer algebra system is recommended.)
x' = −16x + y + x2,     y' = 
9
2
α − y
(a) Sketch the nullclines. (Let α1 < α2 < α3.)


Describe how the critical points move as α increases.
    

(b) Find the critical points.
c1    (x, y) = 
(smaller x-value)
c2    (x, y) = 
(larger x-value)

(c) Let
α = 14.
 Classify each critical point by investigating the corresponding approximate linear system.
c1    
c2    


(d) Find the bifurcation point
α0
 at which the critical points coincide.
α0 =


Locate this critical point.
(x, y) = 


Find the eigenvalues of the approximate linear system. (Enter your answers as a comma-separated list.)
r =

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