Fundamentals of Differential Equations (9th Edition)
9th Edition
ISBN: 9780321977069
Author: R. Kent Nagle, Edward B. Saff, Arthur David Snider
Publisher: PEARSON
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1. [10 points] Given y₁(x) = x²² is a solution to the differential equation x²y"+6xy'+6y=0 (x>0), find a
second linearly independent solution using reduction of order.
>tt 1:32
> trend.1m 1m (sales
> summary(trend.1m)
-
tt) #3###23 (i) ####
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2107.220
57.997 36.332e-16 ***
tt
-43.500
3.067 -14.18 7.72e-15 ***
> trend = ts (fitted (trend.1m), start-start (sales), freq-frequency (sales))
sales trend ###23%23 (ii) ####
as.numeric((1:32 %% 4)
> X
> q1
> q2
> q3
> 94
=
=
=
=
-
as.numeric((1:32 %% 4)
as.numeric((1:32 %% 4)
as.numeric((1:32 %% 4)
== 1)
2)
==
== 3)
==
0)
> season.lm = 1m (resid (trend.1m) 0+q1 + q2 + q3 + q4) #3##23%23 (iii) ####
> summary(season.1m)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
q1
-38.41
43.27 -0.888 0.38232
92
18.80
43.27
0.435 0.66719
q3
-134.78
43.27
-3.115 0.00422 **
94
154.38
43.27 3.568
0.00132 **
> season = ts (fitted (season.lm), start=start (sales), freq=frequency (sales))
> Y X season %23%23%23%23 (iv) ####
>ar (Y, aic=FALSE, order.max=1) #23%23%23%23 (v) ####
Coefficients:
1
0.5704
Order selected 1 sigma 2 estimated as 9431
> ar(Y, aic=FALSE,…
Refer to page 52 for solving the heat equation using separation of variables.
Instructions:
•
•
•
Write the heat equation in its standard form and apply boundary and initial conditions.
Use the method of separation of variables to derive the solution.
Clearly show the derivation of eigenfunctions and coefficients. Provide a detailed solution, step-
by-step.
Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qo Hazb9tC440AZF/view?usp=sharing]
Chapter 4 Solutions
Fundamentals of Differential Equations (9th Edition)
Ch. 4.1 - Verify that for b = 0 and Fext(t) = 0, equation...Ch. 4.1 - If Fext(t) = 0, equation (3) becomes my + by + ky...Ch. 4.1 - Show that if Fext(t) = 0, m = 1, k = 9, and b = 6,...Ch. 4.1 - Verify that y = sin 3t + 2 cos 3t is a solution to...Ch. 4.1 - Prob. 5ECh. 4.1 - An external force F(t) = 2 cos 2t is applied to a...Ch. 4.1 - Prob. 7ECh. 4.1 - In Problems 79, find a synchronous solution of the...Ch. 4.1 - In Problems 79, find a synchronous solution of the...Ch. 4.1 - Prob. 10E
Ch. 4.2 - In Problems 112, find a general solution to the...Ch. 4.2 - In Problems 112, find a general solution to the...Ch. 4.2 - Prob. 3ECh. 4.2 - In Problems 112, find a general solution to the...Ch. 4.2 - Prob. 5ECh. 4.2 - Prob. 6ECh. 4.2 - Prob. 7ECh. 4.2 - Prob. 8ECh. 4.2 - Prob. 9ECh. 4.2 - Prob. 10ECh. 4.2 - In Problems 112, find a general solution to the...Ch. 4.2 - In Problems 112, find a general solution to the...Ch. 4.2 - In Problems 1320, solve the given initial value...Ch. 4.2 - In Problems 1320, solve the given initial value...Ch. 4.2 - Prob. 15ECh. 4.2 - In Problems 1320, solve the given initial value...Ch. 4.2 - Prob. 17ECh. 4.2 - Prob. 18ECh. 4.2 - Prob. 19ECh. 4.2 - In Problems 1320, solve the given initial value...Ch. 4.2 - Prob. 21ECh. 4.2 - Prob. 22ECh. 4.2 - Prob. 23ECh. 4.2 - Prob. 24ECh. 4.2 - Prob. 25ECh. 4.2 - Prob. 26ECh. 4.2 - In Problems 2732, use Definition 1 to determine...Ch. 4.2 - In Problems 2732, use Definition 1 to determine...Ch. 4.2 - In Problems 2732, use Definition 1 to determine...Ch. 4.2 - In Problems 2732, use Definition 1 to determine...Ch. 4.2 - Prob. 31ECh. 4.2 - Prob. 32ECh. 4.2 - Prob. 33ECh. 4.2 - Prob. 34ECh. 4.2 - Prob. 35ECh. 4.2 - Using the definition in Problem 35, prove that if...Ch. 4.2 - Prob. 37ECh. 4.2 - Prob. 38ECh. 4.2 - Prob. 39ECh. 4.2 - Prob. 40ECh. 4.2 - In Problems 3741, find three linearly independent...Ch. 4.2 - Prob. 42ECh. 4.2 - Prob. 43ECh. 4.2 - Solve the initial value problem: y 2y y + 2y =...Ch. 4.2 - Prob. 45ECh. 4.2 - Prob. 46ECh. 4.3 - In Problems 18, the auxiliary equation for the...Ch. 4.3 - In Problems 18, the auxiliary equation for the...Ch. 4.3 - In Problems 18, the auxiliary equation for the...Ch. 4.3 - In Problems 18, the auxiliary equation for the...Ch. 4.3 - Prob. 5ECh. 4.3 - Prob. 6ECh. 4.3 - In Problems 18, the auxiliary equation for the...Ch. 4.3 - In Problems 18, the auxiliary equation for the...Ch. 4.3 - In Problems 920, find a general solution. 9. y 8y...Ch. 4.3 - In Problems 920, find a general solution. 10. y +...Ch. 4.3 - Prob. 11ECh. 4.3 - Prob. 12ECh. 4.3 - In Problems 920, find a general solution. 13. y ...Ch. 4.3 - Prob. 14ECh. 4.3 - Prob. 15ECh. 4.3 - Prob. 16ECh. 4.3 - In Problems 920, find a general solution. 17. y y...Ch. 4.3 - In Problems 920, find a general solution. 18. 2y +...Ch. 4.3 - Prob. 19ECh. 4.3 - In Problems 920, find a general solution. 20. y y...Ch. 4.3 - In Problems 2127, solve the given initial value...Ch. 4.3 - Prob. 22ECh. 4.3 - Prob. 23ECh. 4.3 - In Problems 2127, solve the given initial value...Ch. 4.3 - Prob. 25ECh. 4.3 - In Problems 2127, solve the given initial value...Ch. 4.3 - Prob. 27ECh. 4.3 - Prob. 28ECh. 4.3 - Prob. 29ECh. 4.3 - Prob. 30ECh. 4.3 - Prob. 31ECh. 4.3 - Prob. 32ECh. 4.3 - Prob. 33ECh. 4.3 - Prob. 34ECh. 4.3 - Swinging Door. The motion of a swinging door with...Ch. 4.3 - Prob. 36ECh. 4.3 - Prob. 37ECh. 4.3 - Prove the sum of angles formula for the sine...Ch. 4.4 - In Problems 18, decide whether or not the method...Ch. 4.4 - Prob. 2ECh. 4.4 - Prob. 3ECh. 4.4 - Prob. 4ECh. 4.4 - Prob. 5ECh. 4.4 - Prob. 6ECh. 4.4 - Prob. 7ECh. 4.4 - Prob. 8ECh. 4.4 - In Problems 926, find a particular solution to the...Ch. 4.4 - In Problems 926, find a particular solution to the...Ch. 4.4 - In Problems 926, find a particular solution to the...Ch. 4.4 - In Problems 926, find a particular solution to the...Ch. 4.4 - In Problems 926, find a particular solution to the...Ch. 4.4 - In Problems 926, find a particular solution to the...Ch. 4.4 - In Problems 926, find a particular solution to the...Ch. 4.4 - Prob. 16ECh. 4.4 - In Problems 926, find a particular solution to the...Ch. 4.4 - Prob. 18ECh. 4.4 - Prob. 19ECh. 4.4 - In Problems 926, find a particular solution to the...Ch. 4.4 - In Problems 926, find a particular solution to the...Ch. 4.4 - In Problems 926, find a particular solution to the...Ch. 4.4 - Prob. 23ECh. 4.4 - Prob. 24ECh. 4.4 - Prob. 25ECh. 4.4 - Prob. 26ECh. 4.4 - Prob. 27ECh. 4.4 - Prob. 28ECh. 4.4 - Prob. 29ECh. 4.4 - Prob. 30ECh. 4.4 - Prob. 31ECh. 4.4 - Prob. 32ECh. 4.4 - Prob. 33ECh. 4.4 - Prob. 34ECh. 4.4 - Prob. 35ECh. 4.4 - In Problems 3336, use the method of undetermined...Ch. 4.5 - Given that y1(t) = cos t is a solution to y y + y...Ch. 4.5 - Prob. 2ECh. 4.5 - Prob. 3ECh. 4.5 - Prob. 4ECh. 4.5 - Prob. 5ECh. 4.5 - Prob. 6ECh. 4.5 - Prob. 7ECh. 4.5 - In Problems 38, a nonhomogeneous equation and a...Ch. 4.5 - Prob. 9ECh. 4.5 - Prob. 10ECh. 4.5 - Prob. 11ECh. 4.5 - Prob. 12ECh. 4.5 - Prob. 13ECh. 4.5 - In Problems 916 decide whether the method of...Ch. 4.5 - Prob. 15ECh. 4.5 - Prob. 16ECh. 4.5 - Prob. 17ECh. 4.5 - Prob. 18ECh. 4.5 - Prob. 19ECh. 4.5 - In Problems 1722, find a general solution to the...Ch. 4.5 - Prob. 21ECh. 4.5 - Prob. 22ECh. 4.5 - Prob. 23ECh. 4.5 - Prob. 24ECh. 4.5 - Prob. 25ECh. 4.5 - Prob. 26ECh. 4.5 - Prob. 27ECh. 4.5 - Prob. 28ECh. 4.5 - Prob. 29ECh. 4.5 - Prob. 30ECh. 4.5 - In Problems 3136, determine the form of a...Ch. 4.5 - In Problems 3136, determine the form of a...Ch. 4.5 - Prob. 33ECh. 4.5 - In Problems 3136, determine the form of a...Ch. 4.5 - Prob. 35ECh. 4.5 - Prob. 36ECh. 4.5 - Prob. 37ECh. 4.5 - Prob. 38ECh. 4.5 - In Problems 3740, find a particular solution to...Ch. 4.5 - Prob. 40ECh. 4.5 - Prob. 41ECh. 4.5 - Prob. 42ECh. 4.5 - Prob. 43ECh. 4.5 - Prob. 44ECh. 4.5 - Prob. 46ECh. 4.5 - Prob. 47ECh. 4.5 - Prob. 48ECh. 4.6 - Prob. 1ECh. 4.6 - In Exercises 12, rectangles have been drawn to...Ch. 4.6 - Prob. 3ECh. 4.6 - Prob. 4ECh. 4.6 - Prob. 5ECh. 4.6 - Prob. 6ECh. 4.6 - Prob. 7ECh. 4.6 - Prob. 8ECh. 4.6 - Prob. 9ECh. 4.6 - Prob. 10ECh. 4.6 - Prob. 11ECh. 4.6 - Prob. 12ECh. 4.6 - Prob. 13ECh. 4.6 - Prob. 14ECh. 4.6 - Prob. 15ECh. 4.6 - In Exercises 1318, use a calculator or a computer...Ch. 4.6 - Prob. 17ECh. 4.6 - In Exercises 1318, use a calculator or a computer...Ch. 4.6 - For f(x) = 3x + 2, evaluate the Riemann sums: (a)...Ch. 4.6 - For g(x) = 4x 1, evaluate the Riemann sums: (a)...Ch. 4.6 - Prob. 21ECh. 4.6 - Use Table 5.9 to evaluate the Riemann sums: Table...Ch. 4.6 - The graph of f(t) is in Figure 5.40. Which of the...Ch. 4.6 - Prob. 24ECh. 4.6 - Prob. 25ECh. 4.7 - In Problems 1 through 4, use Theorem 5 to discuss...Ch. 4.7 - In Problems 1 through 4, use Theorem 5 to discuss...Ch. 4.7 - Prob. 3ECh. 4.7 - In Problems 1 through 4, use Theorem 5 to discuss...Ch. 4.7 - Prob. 5ECh. 4.7 - In Problems 5 through 8, determine whether Theorem...Ch. 4.7 - Prob. 7ECh. 4.7 - In Problems 5 through 8, determine whether Theorem...Ch. 4.7 - Prob. 9ECh. 4.7 - In Problems 9 through 14, find a general solution...Ch. 4.7 - Prob. 11ECh. 4.7 - In Problems 9 through 14, find a general solution...Ch. 4.7 - Prob. 13ECh. 4.7 - In Problems 9 through 14, find a general solution...Ch. 4.7 - In Problems 15 through 18, find a general solution...Ch. 4.7 - In Problems 15 through 18, find a general solution...Ch. 4.7 - Prob. 17ECh. 4.7 - Prob. 18ECh. 4.7 - Prob. 19ECh. 4.7 - Prob. 20ECh. 4.7 - Prob. 21ECh. 4.7 - Prob. 22ECh. 4.7 - Prob. 23ECh. 4.7 - Solve the following CauchyEuler equations by using...Ch. 4.7 - Prob. 25ECh. 4.7 - Prob. 26ECh. 4.7 - Prob. 27ECh. 4.7 - Prob. 28ECh. 4.7 - Prove that if y1 and y2 are linearly independent...Ch. 4.7 - Prob. 30ECh. 4.7 - Prob. 31ECh. 4.7 - Prob. 32ECh. 4.7 - Prob. 33ECh. 4.7 - Prob. 34ECh. 4.7 - Prob. 35ECh. 4.7 - Prob. 36ECh. 4.7 - Prob. 37ECh. 4.7 - Prob. 38ECh. 4.7 - Prob. 39ECh. 4.7 - Prob. 40ECh. 4.7 - Prob. 41ECh. 4.7 - In Problems 41 through 44, a differential equation...Ch. 4.7 - Prob. 43ECh. 4.7 - Prob. 44ECh. 4.7 - Prob. 45ECh. 4.7 - Prob. 46ECh. 4.7 - Prob. 47ECh. 4.7 - Prob. 48ECh. 4.7 - Prob. 49ECh. 4.7 - Prob. 50ECh. 4.7 - Prob. 51ECh. 4.7 - Prob. 52ECh. 4.8 - Prob. 1ECh. 4.8 - Prob. 2ECh. 4.8 - Try to predict the qualitative features of the...Ch. 4.8 - Prob. 4ECh. 4.8 - Prob. 5ECh. 4.8 - Prob. 6ECh. 4.8 - Prob. 7ECh. 4.8 - Prob. 8ECh. 4.8 - Prob. 9ECh. 4.8 - Prob. 10ECh. 4.8 - Prob. 11ECh. 4.8 - Use reduction of order to derive the solution...Ch. 4.8 - Prob. 13ECh. 4.8 - Prob. 14ECh. 4.8 - Prob. 15ECh. 4.8 - Prob. 16ECh. 4.8 - Prob. 17ECh. 4.9 - Prob. 1ECh. 4.9 - All problems refer to the massspring configuration...Ch. 4.9 - Prob. 3ECh. 4.9 - Prob. 4ECh. 4.9 - Prob. 5ECh. 4.9 - Prob. 6ECh. 4.9 - Prob. 7ECh. 4.9 - Prob. 8ECh. 4.9 - All problems refer to the massspring configuration...Ch. 4.9 - Prob. 10ECh. 4.9 - Prob. 11ECh. 4.9 - Prob. 12ECh. 4.9 - Prob. 13ECh. 4.9 - Prob. 14ECh. 4.9 - Prob. 15ECh. 4.9 - Prob. 16ECh. 4.9 - Prob. 17ECh. 4.9 - Prob. 18ECh. 4.10 - In the following problems, take g = 32 ft/sec2 for...Ch. 4.10 - Prob. 2ECh. 4.10 - Prob. 3ECh. 4.10 - Prob. 4ECh. 4.10 - Prob. 5ECh. 4.10 - Prob. 6ECh. 4.10 - Prob. 7ECh. 4.10 - Prob. 8ECh. 4.10 - In the following problems, take g = 32 ft/sec2 for...Ch. 4.10 - Prob. 10ECh. 4.10 - Prob. 11ECh. 4.10 - Prob. 12ECh. 4.10 - In the following problems, take g = 32 ft/sec2 for...Ch. 4.10 - Prob. 14ECh. 4.10 - Prob. 15ECh. 4 - In Problems 128, find a general solution to the...Ch. 4 - Prob. 2RPCh. 4 - Prob. 3RPCh. 4 - Prob. 4RPCh. 4 - Prob. 5RPCh. 4 - Prob. 6RPCh. 4 - Prob. 7RPCh. 4 - Prob. 8RPCh. 4 - Prob. 9RPCh. 4 - Prob. 10RPCh. 4 - Prob. 11RPCh. 4 - Prob. 12RPCh. 4 - Prob. 13RPCh. 4 - Prob. 14RPCh. 4 - Prob. 15RPCh. 4 - Prob. 16RPCh. 4 - Prob. 17RPCh. 4 - In Problems 128, find a general solution to the...Ch. 4 - Prob. 19RPCh. 4 - Prob. 20RPCh. 4 - Prob. 21RPCh. 4 - Prob. 22RPCh. 4 - Prob. 23RPCh. 4 - Prob. 24RPCh. 4 - Prob. 25RPCh. 4 - Prob. 26RPCh. 4 - Prob. 27RPCh. 4 - Prob. 28RPCh. 4 - Prob. 29RPCh. 4 - Prob. 30RPCh. 4 - Prob. 31RPCh. 4 - Prob. 32RPCh. 4 - Prob. 33RPCh. 4 - Prob. 34RPCh. 4 - Prob. 35RPCh. 4 - Prob. 36RPCh. 4 - Prob. 37RPCh. 4 - Prob. 38RPCh. 4 - A 32-lb weight is attached to a vertical spring,...Ch. 4 - Prob. 1TWECh. 4 - Prob. 2TWECh. 4 - Prob. 3TWECh. 4 - For students with a background in linear algebra:...
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