Elementary Differential Equations
10th Edition
ISBN: 9780470458327
Author: William E. Boyce, Richard C. DiPrima
Publisher: Wiley, John & Sons, Incorporated
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Chapter 3.4, Problem 4P
To determine
The general solution of the
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4. Consider Chebychev's equation
(1 - x²)y" - xy + λy = 0
with boundary conditions y(-1) = 0 and y(1) = 0, where X is a constant.
(a) Show that Chebychev's equation can be expressed in Sturm-Liouville form
d
· (py') + qy + Ary = 0,
dx
y(1) = 0, y(-1) = 0,
where p(x) = (1 = x²) 1/2, q(x) = 0 and r(x) = (1 − x²)-1/2
(b) Show that the eigenfunctions of the Sturm-Liouville equation are extremals of the
functional A[y], where
A[y]
=
I[y]
J[y]'
and I[y] and [y] are defined by
-
I [y] = √, (my² — qy²) dx
and
J[y] = [[", ry² dx.
Explain briefly how to use this to obtain estimates of the smallest eigenvalue >1.
1
(c) Let k > be a parameter. Explain why the functions y(x) = (1-x²) are suitable
4
trial functions for estimating the smallest eigenvalue. Show that the value of A[y]
for these trial functions is
4k2
A[y] =
=
4k - 1'
and use this to estimate the smallest eigenvalue \1.
Hint:
L₁ x²(1 − ²)³¹ dr =
1
(1 - x²)³ dx
(ẞ > 0).
2ẞ
You recieve a case of fresh Michigan cherries that weighs 8.2 kg. You will be making cherry pies. Each pie will require 1 3/4 pounds of pitted cherries. How many pies can be made from the case if the yield percent for cherries is 87
Chapter 3 Solutions
Elementary Differential Equations
Ch. 3.1 - In each of Problems 1 through 8, find the general...Ch. 3.1 - In each of Problems 1 through 8, find the general...Ch. 3.1 - In each of Problems 1 through 8, find the general...Ch. 3.1 - In each of Problems 1 through 8, find the general...Ch. 3.1 - In each of Problems 1 through 8, find the general...Ch. 3.1 - In each of Problems 1 through 8, find the general...Ch. 3.1 - In each of Problems 1 through 8, find the general...Ch. 3.1 - In each of Problems 1 through 8, find the general...Ch. 3.1 - In each of Problems 9 through 16, find the...Ch. 3.1 - In each of Problems 9 through 16, find the...
Ch. 3.1 - In each of Problems 9 through 16, find the...Ch. 3.1 - Prob. 12PCh. 3.1 - Prob. 13PCh. 3.1 - Prob. 14PCh. 3.1 - In each of Problems 9 through 16, find the...Ch. 3.1 - Prob. 16PCh. 3.1 - Prob. 17PCh. 3.1 - Prob. 18PCh. 3.1 - Prob. 19PCh. 3.1 - Prob. 20PCh. 3.1 - Solve the initial value problem y″ − y′ − 2y = 0,...Ch. 3.1 - Solve the initial value problem 4y″ − y = 0, y(0)...Ch. 3.1 - Prob. 23PCh. 3.1 - Prob. 24PCh. 3.1 - Prob. 25PCh. 3.1 - Prob. 26PCh. 3.1 - Prob. 27PCh. 3.1 - Prob. 28PCh. 3.2 - In each of Problems 1 through 6, find the...Ch. 3.2 - In each of Problems 1 through 6, find the...Ch. 3.2 - In each of Problems 1 through 6, find the...Ch. 3.2 - In each of Problems 1 through 6, find the...Ch. 3.2 - In each of Problems 1 through 6, find the...Ch. 3.2 - In each of Problems 1 through 6, find the...Ch. 3.2 - In each of Problems 7 through 12, determine the...Ch. 3.2 - In each of Problems 7 through 12, determine the...Ch. 3.2 - In each of Problems 7 through 12, determine the...Ch. 3.2 - In each of Problems 7 through 12, determine the...Ch. 3.2 - In each of Problems 7 through 12, determine the...Ch. 3.2 - In each of Problems 7 through 12, determine the...Ch. 3.2 - Verify that y1(t) = t2 and y2(t) = t−1 are two...Ch. 3.2 - Verify that y1(t) = 1 and y2(t) = t1/2 are...Ch. 3.2 - Show that if y = φ(t) is a solution of the...Ch. 3.2 - Can y = sin(t2) be a solution on an interval...Ch. 3.2 - If the Wronskian W of f and g is 3e4t, and if f(t)...Ch. 3.2 - Prob. 18PCh. 3.2 - If W(f, g) is the Wronskian of f and g, and if u =...Ch. 3.2 - If the Wronskian of f and g is t cos t − sin t,...Ch. 3.2 - Assume that y1 and y2 are a fundamental set of...Ch. 3.2 - Prob. 22PCh. 3.2 - Prob. 23PCh. 3.2 - Prob. 24PCh. 3.2 - Prob. 25PCh. 3.2 - Prob. 26PCh. 3.2 - Prob. 27PCh. 3.2 - Prob. 28PCh. 3.2 - Prob. 29PCh. 3.2 - Prob. 30PCh. 3.2 - Prob. 31PCh. 3.2 - Prob. 32PCh. 3.2 - Prob. 33PCh. 3.2 - Prob. 34PCh. 3.2 - Prob. 35PCh. 3.2 - If the Wronskian of any two solutions of y″ +...Ch. 3.2 - Prob. 37PCh. 3.2 - Prob. 38PCh. 3.2 - Prob. 39PCh. 3.2 - Prob. 40PCh. 3.2 - Prob. 41PCh. 3.2 - Prob. 42PCh. 3.2 - Prob. 43PCh. 3.2 - Prob. 44PCh. 3.2 - Prob. 45PCh. 3.2 - Prob. 46PCh. 3.2 - Prob. 47PCh. 3.2 - Prob. 48PCh. 3.2 - Prob. 49PCh. 3.2 - Prob. 50PCh. 3.2 - Prob. 51PCh. 3.3 - In each of Problems 1 through 6, use Euler’s...Ch. 3.3 - In each of Problems 1 through 6, use Euler’s...Ch. 3.3 - In each of Problems 1 through 6, use Euler’s...Ch. 3.3 - In each of Problems 1 through 6, use Euler’s...Ch. 3.3 - In each of Problems 1 through 6, use Euler’s...Ch. 3.3 - In each of Problems 1 through 6, use Euler’s...Ch. 3.3 - In each of Problems 7 through 16, find the general...Ch. 3.3 - In each of Problems 7 through 16, find the general...Ch. 3.3 - In each of Problems 7 through 16, find the general...Ch. 3.3 - In each of Problems 7 through 16, find the general...Ch. 3.3 - Prob. 11PCh. 3.3 - Prob. 12PCh. 3.3 - In each of Problems 7 through 16, find the general...Ch. 3.3 - Prob. 14PCh. 3.3 - Prob. 15PCh. 3.3 - Prob. 16PCh. 3.3 - Prob. 17PCh. 3.3 - Prob. 18PCh. 3.3 - Prob. 19PCh. 3.3 - Prob. 20PCh. 3.3 - In each of Problems 17 through 22, find the...Ch. 3.3 - In each of Problems 17 through 22, find the...Ch. 3.3 - Prob. 23PCh. 3.3 - Prob. 24PCh. 3.3 - Prob. 25PCh. 3.3 - Prob. 26PCh. 3.3 - Prob. 27PCh. 3.3 - Prob. 28PCh. 3.3 - Prob. 29PCh. 3.3 - Prob. 30PCh. 3.3 - Prob. 31PCh. 3.3 - Prob. 32PCh. 3.3 - Prob. 33PCh. 3.3 - Prob. 34PCh. 3.3 - Prob. 35PCh. 3.3 - Prob. 36PCh. 3.3 - Prob. 37PCh. 3.3 - Prob. 38PCh. 3.3 - Prob. 39PCh. 3.3 - Prob. 40PCh. 3.3 - Prob. 41PCh. 3.3 - Prob. 42PCh. 3.3 - Prob. 43PCh. 3.3 - Prob. 44PCh. 3.3 - Prob. 45PCh. 3.3 - Prob. 46PCh. 3.4 - In each of Problems 1 through 10, find the general...Ch. 3.4 - Prob. 2PCh. 3.4 - Prob. 3PCh. 3.4 - Prob. 4PCh. 3.4 - Prob. 5PCh. 3.4 - Prob. 6PCh. 3.4 - Prob. 7PCh. 3.4 - Prob. 8PCh. 3.4 - Prob. 9PCh. 3.4 - Prob. 10PCh. 3.4 - In each of Problems 11 through 14, solve the given...Ch. 3.4 - Prob. 12PCh. 3.4 - Prob. 13PCh. 3.4 - Prob. 14PCh. 3.4 - Prob. 15PCh. 3.4 - Prob. 16PCh. 3.4 - Prob. 17PCh. 3.4 - Consider the initial value problem
9y″ + 12y′ + 4y...Ch. 3.4 - Prob. 19PCh. 3.4 - Prob. 20PCh. 3.4 - Prob. 21PCh. 3.4 - Prob. 22PCh. 3.4 - Prob. 23PCh. 3.4 - Prob. 24PCh. 3.4 - Prob. 25PCh. 3.4 - Prob. 26PCh. 3.4 - Prob. 27PCh. 3.4 - Prob. 28PCh. 3.4 - Prob. 29PCh. 3.4 - Prob. 30PCh. 3.4 - Prob. 31PCh. 3.4 - The method of Problem 20 can be extended to second...Ch. 3.4 - In each of Problems 33 through 36, use the method...Ch. 3.4 - Prob. 34PCh. 3.4 - Prob. 35PCh. 3.4 - Prob. 36PCh. 3.4 - Prob. 37PCh. 3.4 - Prob. 38PCh. 3.4 - Prob. 39PCh. 3.4 - Euler Equations. In each of Problems 40 through...Ch. 3.4 - Prob. 41PCh. 3.4 - Prob. 42PCh. 3.4 - Prob. 43PCh. 3.4 - Prob. 44PCh. 3.4 - Prob. 45PCh. 3.5 - In each of Problems 1 through 14, find the general...Ch. 3.5 - In each of Problems 1 through 14, find the general...Ch. 3.5 - In each of Problems 1 through 14, find the general...Ch. 3.5 - Prob. 4PCh. 3.5 - Prob. 5PCh. 3.5 - Prob. 6PCh. 3.5 - Prob. 7PCh. 3.5 - In each of Problems 1 through 14, find the general...Ch. 3.5 - Prob. 9PCh. 3.5 - Prob. 10PCh. 3.5 - In each of Problems 1 through 14, find the general...Ch. 3.5 - Prob. 12PCh. 3.5 - Prob. 13PCh. 3.5 - Prob. 14PCh. 3.5 - Prob. 15PCh. 3.5 - Prob. 16PCh. 3.5 - In each of Problems 15 through 20, find the...Ch. 3.5 - Prob. 18PCh. 3.5 - Prob. 19PCh. 3.5 - Prob. 20PCh. 3.5 - Prob. 29PCh. 3.5 - Prob. 30PCh. 3.5 - Prob. 31PCh. 3.5 - Prob. 32PCh. 3.5 - Prob. 33PCh. 3.5 - Prob. 34PCh. 3.5 - Prob. 35PCh. 3.5 - Prob. 36PCh. 3.5 - Prob. 37PCh. 3.5 - Prob. 38PCh. 3.5 - Prob. 39PCh. 3.6 - In each of Problems 1 through 4, use the method of...Ch. 3.6 - Prob. 2PCh. 3.6 - Prob. 3PCh. 3.6 - Prob. 4PCh. 3.6 - Prob. 5PCh. 3.6 - Prob. 6PCh. 3.6 - Prob. 7PCh. 3.6 - Prob. 8PCh. 3.6 - Prob. 9PCh. 3.6 - Prob. 10PCh. 3.6 - Prob. 11PCh. 3.6 - Prob. 12PCh. 3.6 - Prob. 13PCh. 3.6 - Prob. 14PCh. 3.6 - Prob. 15PCh. 3.6 - Prob. 16PCh. 3.6 - Prob. 17PCh. 3.6 - Prob. 18PCh. 3.6 - Prob. 19PCh. 3.6 - Prob. 20PCh. 3.6 - Prob. 21PCh. 3.6 - Prob. 22PCh. 3.6 - Prob. 23PCh. 3.6 - Prob. 24PCh. 3.6 - Prob. 25PCh. 3.6 - Prob. 26PCh. 3.6 - Prob. 27PCh. 3.6 - Prob. 28PCh. 3.6 - Prob. 29PCh. 3.6 - Prob. 30PCh. 3.6 - Prob. 31PCh. 3.6 - Prob. 32PCh. 3.7 - In each of Problems 1 through 4, determine ω0, R,...Ch. 3.7 - Prob. 2PCh. 3.7 - Prob. 3PCh. 3.7 - Prob. 4PCh. 3.7 - Prob. 5PCh. 3.7 - Prob. 6PCh. 3.7 - Prob. 7PCh. 3.7 - Prob. 8PCh. 3.7 - Prob. 9PCh. 3.7 - Prob. 10PCh. 3.7 - Prob. 11PCh. 3.7 - Prob. 12PCh. 3.7 - Prob. 13PCh. 3.7 - Prob. 14PCh. 3.7 - Prob. 15PCh. 3.7 - Prob. 16PCh. 3.7 - Prob. 17PCh. 3.7 - Prob. 18PCh. 3.7 - Prob. 19PCh. 3.7 - Prob. 20PCh. 3.7 - Prob. 21PCh. 3.7 - Prob. 22PCh. 3.7 - Prob. 23PCh. 3.7 - Prob. 24PCh. 3.7 - Prob. 26PCh. 3.7 - Prob. 27PCh. 3.7 - Prob. 28PCh. 3.7 - Prob. 29PCh. 3.7 - Prob. 30PCh. 3.7 - Prob. 31PCh. 3.8 - Prob. 1PCh. 3.8 - Prob. 2PCh. 3.8 - Prob. 3PCh. 3.8 - Prob. 4PCh. 3.8 - Prob. 5PCh. 3.8 - Prob. 6PCh. 3.8 - Prob. 7PCh. 3.8 - Prob. 8PCh. 3.8 - Prob. 9PCh. 3.8 - Prob. 10PCh. 3.8 - Prob. 11PCh. 3.8 - Prob. 12PCh. 3.8 - Prob. 14PCh. 3.8 - Prob. 15PCh. 3.8 - Prob. 16P
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- A: Show that the ODES has no limit cycle in a region D and find this region: x=y-2x³ y=x+y-2y3 Carrow_forwardoptımızatıon theoryarrow_forwardQ3)A: Given H(x,y)= x²-x4 + y² as a first integral of an ODEs, find this ODES corresponding to H(x,y) and show the phase portrait by using Hartman theorem and by drawing graph of H(x,y)=c. Discuss the stability of critical points of the corresponding ODEs.arrow_forward
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