For a power series solution about x=0 for a IVP. The initialconditions are y(0)=8 and y'(0)= -15. The following recurrencerelation was obtained.Can+2 = (−1)nnn+ 2∙AnFind the first six terms of the power series solution for this IVP

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Answered: For a power series solution about x=0…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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The Legendre’s differential equation is (1 − x2)y'' − 2xy' + l(l + 1)y = 0, where l is a constant, and y = y(x). Consider a series solution about the point x = 0 of the form provided, where k is a constant and the an are coefficients that need to be determined. Show that the recurrence relation is given by an+2 (n + k + 2)(n + k + 1) = an [(n + k)(n + k + 1) − l(l + 1)].
Given that y' = yx2 + 2x, where y = -1 at x = 0, = 0, %3D and h=0.2. Estimate the values of y at x = 0.4 and 0.6 using this recurrence relation.
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(a) Seek power series solutions of Airy differential equation y" – xy = 0 about x = 0; find the recurrence relation. (b) Find the first four terms in each of two solutions y1 and y2.
In solving a differential equation by Frobenius method of series solution about x = 0, we obtain the following indicial equation and recurrence relation: 7(2r–1)=0 and 1 Cn+1 = %3D 6n +r+3n where n = 0,1,2,3,... Find the solution corresponding to the larger root of the indicial equation. Include the first three nonzero terms, use Co=I O A. Y = 1+x 4 x+... 133 O B. Y =x-1/2/ 4 3 +.. 133 O c.y =xl/2| 12 4 3 +... c. 4 2 O D. y = x1/2 1+x+ 133 ++.. O E.Y =x1/2 1+x+ 19 4 2 +...
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1.) Given xy^''+2y^'-y=0 solve by the method of Frobenius when r=1 and conclude whether r=1 is correct value or can be used to solve the differential equation. After identifying the first three terms, which will become the leader of the REMAINING SERIES? Show how the 2 other series will become a member of the REMAINING SERIES Calculate the RECURRENCE FORMULA: the values of a3, a4, & a5 Conclusion: Is the proper value obtained from the indicial equation? Yes or no, briefly explain.
Solve the given ODE using the power series method. y'' + xy' + 2y = 0 4. Obtain the recurrence relation. as Oas+2 = s + 1 as O as+2 s + 2 as as+2 = 8 + 2 as O as+2 = - s + 1 5. Determine the general solution. Use up to the 5th-degree term of the solution. 1 1 + a1 x 3 1 ,5 + 15 Y = ao[ 1 - - (-- 1 1 x° + 2 y = ao 1 + a1 - ... 1 y = ao[ 1 + 2 1 + -x* + 8 1 x° + 15 + ai x + 3 ... ... 1 4 + 1 + a1( x + 1 y = ao(1+x² + + 3 8 - |00
2- Use the method of power series to solve the following ODE. Write the recurrence formula and write the first 8 terms of the series. Write the general solution if possible (2+x²)y' - xy + 4y = 0, y(0) = -1, y(0) = 3 about x=0
Solve for the general solution using the Power Series. The answer was already given, you just have to provide the solutions. Find the G.S. about xo = 0 for the following DE's 3. (x² + 4)y" + xy = x + 2 1 8 a₁ +5 2-ao 1 + ² x ² + 1 20400 + ... Answer: y = ·Σa₂x² = a₁ + a₁x + ² x ² 48 320 72=0 ²x4 ts
A3 Find a series expansion, up to x6, for the general solution of Hermite's equation: y" - xy + sy = 0. Write the expansions for the specific solutions y₁, y2, satisfying y₁ (0) = 1,; y₁ (0) = 0 and y2 (0) = 0, ; 3₂(0) = 1. Use the recurrence relation to show that the series for either y₁ or for y2 truncates to a polynomial if s is a positive integer. Write down the explicit forms of the expansions for y₁ and y2 for each of the cases s 3 and s = 4.
Find the first four nonzero terms in a power series expansion of the solution to the given initial value problem. 3y - 2²xy=0; y(0) = 2 y(x)=+ +... (Type an expression that includes all terms up to order 3.)

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For a power series solution about x=0 for a IVP. The initial conditions are y(0)=8 and y'(0)= -15. The following recurrence relation was obtained. C C an+2 = (−1)n. n n+2 an Find the first six terms of the power series solution for this IVP