Explanation Given: Differential equation is ( 4 − x 2 ) y ″ + 2 y = 0 with initial value y ( 0 ) = 0 and y ′ ( 0 ) = 1 . Calculation: From Problem 10, the solution of the differentiable equation ( 4 − x 2 ) y ″ + 2 y = 0 at x 0 = 0 is y = a 0 [ 1 − x 2 4 ] + a 1 [ ∑ n = 0 ∞ x 2 n + 1 4 n ( 2 n − 1 ) ( 2 n + 1 ) ] . That is, y = a 0 [ 1 − x 2 4 ] + a 1 [ x − x 3 12 − x 5 240 − x 7 2240 − x 9 16128 − ⋯ ] Given initial conditions are y ( 0 ) = 0 and y ′ ( 0 ) = 1 . Equate y ( 0 ) = 0 as follows. y ( 0 ) = 0 a 0 [ 1 − ( 0 ) 2 4 ] + a 1 [ 0 − 0 3 12 − ( 0 ) 5 240 − ( 0 ) 7 2240 + ⋯ ] = 0 a 0 ( 1 ) + a 1 ( 0 ) = 0 a 0 = 0 Thus, the value of a 0 is 0. Compute the derivative of y as follows. y ′ ( x ) = ( a 0 [ 1 − x 2 4 ] + a 1 [ x − x 3 12 − x 5 240 − x 7 2240 − x 9 16128 + ⋯ ] ) ′ = a 0 ( 0 − 2 x 4 ) + a 1 ( 1 − 3 x 2 12 − 5 x 4 240 − 7 x 6 2240 − 9 x 8 16128 + … ) = a 0 ( 0 − x 2 ) + a 1 ( 1 − x 2 4 − x 4 48 − x 6 320 − x 8 1792 … ) Equate y ′ ( 0 ) = 1 as follows

10th Edition

ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

See similar textbooks

Videos

Question

Chapter 5.2, Problem 26P

To determine

To sketch: The partial sums in a series of the given initial value problem about x=0.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 5.2, Problem 25P

Chapter 5.2, Problem 27P

Students have asked these similar questions

9.7 Given the equations 0.5x₁-x2=-9.5 1.02x₁ - 2x2 = -18.8 (a) Solve graphically. (b) Compute the determinant. (c) On the basis of (a) and (b), what would you expect regarding the system's condition? (d) Solve by the elimination of unknowns. (e) Solve again, but with a modified slightly to 0.52. Interpret your results.

12.42 The steady-state distribution of temperature on a heated plate can be modeled by the Laplace equation, 0= FT T + 200°C 25°C 25°C T22 0°C T₁ T21 200°C FIGURE P12.42 75°C 75°C 00°C If the plate is represented by a series of nodes (Fig. P12.42), cen- tered finite-divided differences can be substituted for the second derivatives, which results in a system of linear algebraic equations. Use the Gauss-Seidel method to solve for the temperatures of the nodes in Fig. P12.42.

9.22 Develop, debug, and test a program in either a high-level language or a macro language of your choice to solve a system of equations with Gauss-Jordan elimination without partial pivoting. Base the program on the pseudocode from Fig. 9.10. Test the program using the same system as in Prob. 9.18. Compute the total number of flops in your algorithm to verify Eq. 9.37. FIGURE 9.10 Pseudocode to implement the Gauss-Jordan algorithm with- out partial pivoting. SUB GaussJordan(aug, m, n, x) DOFOR k = 1, m d = aug(k, k) DOFOR j = 1, n aug(k, j) = aug(k, j)/d END DO DOFOR 1 = 1, m IF 1 % K THEN d = aug(i, k) DOFOR j = k, n aug(1, j) END DO aug(1, j) - d*aug(k, j) END IF END DO END DO DOFOR k = 1, m x(k) = aug(k, n) END DO END GaussJordan

Chapter 5 Solutions

Elementary Differential Equations

Knowledge Booster

$Advanced Math$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

The partial sums in a series of the given initial value problem about x = 0 .

Videos

To sketch: The partial sums in a series of the given initial value problem about x=0.

Want to see the full answer?

Chapter 5 Solutions