a. Use Power Series to find the solution of the homogeneous DE given by + 3xy = 0.d²ydx²b. Start with the solution y(x)= a₁ + ax + a₂x² + a₂x³ + a²x¹ + açx³ +..., and calculate theexpansions for the first two terms on the LHS of the equation.c. Get the recurrence relations between the a's and find the first 5 terms in the solution.d. Suppose that the initial condition for y(x) is y(0) = 1 and y'(0) = 0. What will the series becomein this case?

Answered: a. Use Power Series to find the…

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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Consider the differential equation x^2y" + x (1−x)y' − xy = 0. a. Show that X0=0 is a regular singular point. b. Determine the indicial equation, the recurrence relation, and the roots of the indicial equation. c. Find the series solution for x>0 corresponding to the larger root. d. Find the series solution corresponding to the smaller root by following the procedure outlined in section 5.4 and demonstrated in section 5.7.
Q.x2(x2+1)y′′+7xexy′+9(1+tanx)y =0
2. Use Taylor series expansions to derive the error term of the sided difference ap- proximation to f" (x0): - f(xo + 2h) + 4f (xo + h) – 3f(xo) Dnf (xo) (1) 2h
Find the recurrence relation to solve the following differential equation about x = 0. (x – 1)y" + xy' = 0 О (п+ 1)(п + 2) ст12 — п(п + 1) Сл+1 — пст — 0, п >1 О (п+1)(п +2) сп+2 — п(п + 1)ст+1 3 0, п>1 о (п+ 1)(n +2)сп12 + п(п + 1)сл11 + псп — 0, п>1 О (n+ 1) (п + 2)сп+2 — псп — 0, п>1 о (п+2)сп+2 — пСп+1 псп — 0, п >1
H is a fixed number , AB=15 f(x) = { 0 -AB < x < 0 0 < x < AB H. Find the period of the function, calculate the coefficients in the Fourier series expansion, and mark which values n=1 are. а. an=0.637H; a0=0;bn=0 O b. an=0; a0=H;bn=0.637H O c. an=2.pi.H/11; a0=H/11;bn=0 O d. an=H; a0=0;bn=H е. an=0; a0=H;bn=0
Find singular points of the differential equation 2x2y′′ − xy′ + (1 + x)y = 0, and solve it around that singular points using power series solutions. Find the first three nonzero terms in each of two solutions about the singular point. Solve the recurrence relation you find (i.e express an in terms of n) and finally show that solutions you find is converging. (Hint: Find the roots of indicial equation and construct the recurrence relation).
2) Suppose you have a differential equation that satisfies the conditions of Theorem 6.2.1- Existence of Power Series Solutions with center x, = 0. It is then guaranteed that a solution exists of the form y(x) = E-0 Cnx". The recurrence relation for the coefficients was calculated to be k 3 Ck+2 k = 1,2,3, .. 2(k + 2) k+1 2(k + 2)(k + 1) 3 C2 = -7C0 Find the solution y(x) to the initial-value problem if y(0) = 0 and y'(0) = 1. Give at least the first 3 nonzero terms of the solution.
2.3.9 Write down the first three terms of the Taylor series expansions of the functions dx (th) and (1-2h) dx dt dt about x(t). Use these two equations to eliminate d²x (t) and d³x dt³ dt from the Taylor series expansion of the function x(t + h) about x(t). Show that the resulting formula for x(t + h) is the third member of the Adams- Bashforth family, and hence confirm that this Adams-Bashforth method is a third-order method. €
Consider the differential equation (x2−1)y′′+[1−(a+b)]xy′+aby=0, (11.7.9) where a and b are constants.
Determine f(-2) and f(4) (-2) for a function with Taylor series T(x) = 3(x + 2) + (x +2)2 – 4(x + 2)3 + 2(x + 2)4 +...
[7] Find the solution around x = 0 of the differential equation (x - 1)y" + xy - y = 0. Find the recurrence relation and keep terms to order 2³.
Find the first five coefficients for this Taylor series C0=? C1=? C2=? C3=? C4=?

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