22. a. Suppose y is a solution of the constant coefficient homogeneous equation Let z(x) = y(x − x0), where x is an arbitrary real number. Show that - ay"+by+cy= 0. az" + bz' +cz = 0. b. Let z₁(x) = y₁ (x − x0) and 22(x) = Y2(x − x0), where {1, 2} is a fundamental set of solutions of (A). Show that {1, 2} is also a fundamental set of solutions of (A). c. The statement of Theorem 5.2.1 is convenient for solving an initial value problem ay" +by+cy = 0, y(0) = ko, y'(0) = k1, where the initial conditions are imposed at x = 0. However, if the initial value problem is ay" +by+cy = 0, y(x0) = ko, y (x0) = k₁, where x = 0, then determining the constants in y= C₁e¹¹² + C₂e¹², y = e¹¹² (c₁ + €2x), or y = e¹² (c₁ cos wx + C₂ sin wx) (whichever is applicable) is more complicated. Use (b) to restate Theorem 5.2.1 in a form more convenient for solving (B). (A) (B)
22. a. Suppose y is a solution of the constant coefficient homogeneous equation Let z(x) = y(x − x0), where x is an arbitrary real number. Show that - ay"+by+cy= 0. az" + bz' +cz = 0. b. Let z₁(x) = y₁ (x − x0) and 22(x) = Y2(x − x0), where {1, 2} is a fundamental set of solutions of (A). Show that {1, 2} is also a fundamental set of solutions of (A). c. The statement of Theorem 5.2.1 is convenient for solving an initial value problem ay" +by+cy = 0, y(0) = ko, y'(0) = k1, where the initial conditions are imposed at x = 0. However, if the initial value problem is ay" +by+cy = 0, y(x0) = ko, y (x0) = k₁, where x = 0, then determining the constants in y= C₁e¹¹² + C₂e¹², y = e¹¹² (c₁ + €2x), or y = e¹² (c₁ cos wx + C₂ sin wx) (whichever is applicable) is more complicated. Use (b) to restate Theorem 5.2.1 in a form more convenient for solving (B). (A) (B)
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.9: Properties Of Determinants
Problem 46E
Related questions
Question
Transcribed Image Text:22.
a. Suppose y is a solution of the constant coefficient homogeneous equation
Let z(x) = y(x − x0), where x is an arbitrary real number. Show that
-
ay"+by+cy= 0.
az" + bz' +cz = 0.
b. Let z₁(x) = y₁ (x − x0) and 22(x) = Y2(x − x0), where {1, 2} is a fundamental set of solutions of (A). Show that {1, 2} is also a fundamental set of
solutions of (A).
c. The statement of Theorem 5.2.1 is convenient for solving an initial value problem
ay" +by+cy = 0, y(0) = ko, y'(0) = k1,
where the initial conditions are imposed at x = 0. However, if the initial value problem is
ay" +by+cy = 0, y(x0) = ko, y (x0) = k₁,
where x = 0, then determining the constants in
y= C₁e¹¹² + C₂e¹², y = e¹¹² (c₁ + €2x), or y = e¹² (c₁ cos wx + C₂ sin wx)
(whichever is applicable) is more complicated. Use (b) to restate Theorem 5.2.1 in a form more convenient for solving (B).
(A)
(B)
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