Explanation The given differential equation is 2 x y ″ − y ′ + 2 y = 0 (1) Consider the solution as y = ∑ n = 0 ∞ c n x n + r . Differentiating y with respect to x gives, y ′ = ∑ n = 0 ∞ ( n + r ) c n x n + r − 1 y ″ = ∑ n = 0 ∞ ( n + r ) ( n + r − 1 ) c n x n + r − 2 Use the values of y , y ′ , y ″ in equation (1) then, 2 x y ″ − y ′ + 2 y = 0 2 x ∑ n = 0 ∞ ( n + r ) ( n + r − 1 ) c n x n + r − 2 − ∑ n = 0 ∞ ( n + r ) c n x n + r − 1 + 2 ∑ n = 0 ∞ c n x n + r = 0 ∑ n = 0 ∞ 2 ( n + r ) ( n + r − 1 ) c n x n + r − 1 − ∑ n = 0 ∞ ( n + r ) c n x n + r − 1 + ∑ n = 0 ∞ 2 c n x n + r = 0 ∑ n = 0 ∞ [ 2 ( n + r ) ( n + r − 1 ) − ( n + r ) ] c n x n + r − 1 + ∑ n = 0 ∞ 2 c n x n + r = 0 Substitute n = k in first summation and n = k − 1 in the second summation, and then simplify ∑ k = 0 ∞ [ 2 ( k + r ) ( k + r − 1 ) − ( k + r ) ] c k x k + r − 1 + ∑ k = 1 ∞ 2 c k − 1 x k + r − 1 = 0 [ 2 ( r ) ( r − 1 ) − r ] c 0 x r − 1 + ∑ k = 1 ∞ [ 2 ( k + r ) ( k + r − 1 ) − ( k + r ) ] c k x k + r − 1 + ∑ k = 1 ∞ 2 c k − 1 x k + r − 1 = 0 [ 2 r 2 − 3 r ] c 0 x r − 1 + ∑ k = 1 ∞ [ 2 ( k + r ) ( k + r − 1 ) c k − ( k + r ) c k + 2 c k − 1 ] x k + r − 1 = 0 Thus, it gives 2 r 2 − 3 r = 0 and 2 ( k + r ) ( k + r − 1 ) c k − ( k + r ) c k + 2 c k − 1 = 0 r ( 2 r − 3 ) = 0 and ( k + r ) ( 2 k + 2 r − 3 ) c k + 2 c k − 1 = 0 On solving it gives the value of indicial roots r = 0 , 3 2

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

11th Edition

ISBN: 9781305965720

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 6.3, Problem 15E

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To show: That the indicial roots do not differ by an integer and then find the linearly independent series solution about x=0.

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Chapter 6.3, Problem 14E

Chapter 6.3, Problem 16E

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(a) (b) (c) (d) de unique? Answer the following questions related to the linear system x + y + z = 2 x-y+z=0 2x + y 2 3 rewrite the linear system into the matrix-vector form A = 5 Fuse elementary row operation to solve this linear system. Is the solution use elementary row operation to find the inverse of A and then solve the linear system. Verify the solution is the same as (b). give the null space of matrix A and find the dimension of null space. give the column space of matrix A and find the dimension of the column space of A (Hint: use Rank-Nullity Theorem).

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[) Hwk 29 SUBMIT ANSWEK Hwk 30 - (MA 244-03) (SP25) || X - Mind Tap Cengage Learning ☑ MA244-03_Syllabus_Spring, 20 × b Answered: [) 90% Hwk 29 Hwk X Rotation of Axes Example - Elimi X + https://www.webassign.net/web/Student/Assignment-Responses/last?dep=36606609 B שי 90% 2. [-/3 Points] DETAILS MY NOTES LARLINALG8 7.4.003. Use the age transition matrix L and age distribution vector X1 to find the age distribution vectors X2 and x3. 0 34 x2 = X3 = L = ↓ ↑ 1 0 0 x1 = 1 0 0 2 20 20 20 Then find a stable age distribution vector. x = t ↓ 1 Need Help? Read It SUBMIT ANSWER 3. [-/3 Points] DETAILS MY NOTES LARLINALG8 7.4.004. Use the age transition matrix L and age distribution vector X1 to find the age distribution vectors x2 and ×3. ill { ASK YOUR TEACHER PRACTICE ANOTHER ASK YOUR TEACHER PRACTICE ANOTHER

Chapter 6 Solutions

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That the indicial roots do not differ by an integer and then find the linearly independent series solution about x = 0 .

Solution Summary: The author shows that the indicial roots do not differ by an integer and then finds the linearly independent series solution about x=0.

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To show: That the indicial roots do not differ by an integer and then find the linearly independent series solution about x=0.

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Chapter 6 Solutions