Question 8Given the differential equation:AB(1+x2) y + 2xy - 2y=0y=c,(1+x arctan(x)) +cThe recurrence relation is given by:3n+27 (₁)11The recurrence relation is given by:1-n1+n+27(-y=carctan(x) +c₁x²1+nn=0,1,2,...n=0,1,2,...

Recurrence relation of differential equation using power series

Answered: Question 8 Given the differential…

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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Solve the following differential equation by series method, x² y" + ( x² + 2 x ) y' – 2 y = 0 If the indicial equation: a, (c – 1)(c + 2) = 0 1 , and recurrence formula: an = аn-1 n+c+2
Can you help me with the third part?
Q1. Simplify (cos-isin ) (cos 70+isin 70) (cos 40+ i sin 40)² (cos+ i sin 0)³ using De Movier's Theorem. Q2. Obtain the Power Series solution of the differential equation y"-xy'+y = 0. Q3. Find the Fourier series for the function: f(x)=[x² for 1
Can you show how exactly to solve these using separation of variables and Fourier Series?
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.
= 1) The function f(x) periodic on the interval [0, 2л] has complex Fourier series f(x): Σ(1/n²) einx where the sum over n goes from - infinity to infinity. Convert this to cosine and sine Fourier Series by finding the values of A's and B's in the expression Ao + ΣAn cos(nx) + Σ Bn sin(nx) where each sum goes from 1 to infinity. Hint: consider the n and -n term together in the complex Fourier Series or use Euler's identity.
Example 7.2. Find a solution of the following system of ODEs using method of variation of parameters. Use y₁ (0): = 19 and y₂ (0) = -23. * = [6²1]+[A]* y' 3 5 5 et
paper 2 4. INDICES AND SURDS 2007 J Q2 (i) Express 1 as a power of 2. V32 (ii) Express (64)* as a power of 2. (iii) Hence solve the equation 2* (64)* V32 1
Find the recurrence relation to solve the following differential equation about z = 0 (z+1)y" + 2zy' = 0 Lütfen birini seçin: O (n +1)(n+2)a,+2 – n(n+1)an+1 – 2na, = 0, for n>1 %3D O (n +1)(n+ 2)an12 – n(n+1)an+1 = 0, for n>1 %3D O (n + 2)an+2 – nan+1 – 2nan = 0, for n21 %3D O (n +1)(n +2)an+2 – 2nan = 0, for n>1 %3| O (n +1)(n + 2)an+2 + n(n+ 1)an+1 +2na, = 0, for n 21 %3D
Provide a coherent answer for items C and D.
HAND WRITTEN WORK ON PAPER ONLY
- Find the Fourier series expansion of the given function, whose definition in one period as: (24) f(x)=cosx (25) f(x) = sinx | (26)f(x)=x| (27) f(x) = sin(tx) 三 (28) f(x)=x3 (29) f (t)= (30) f ( )= sint 一π n-t

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Question 8 Given the differential equation: A y=c,(1+x arctan(x)) +c Ⓡ (1+x2) y + 2xy - 2y=0 The recurrence relation is given by: y=co *+27(- Ca The recurrence relation is given by: *+2=(- arctan(x) +c rẻ 1+n 1-n n -) c 1+n n=0,1,2,... n=0,1,2,...

Question 8 Given the differential equation: A y=c,(1+x arctan(x)) +c Ⓡ (1+x2) y + 2xy - 2y=0 The recurrence relation is given by: y=co +27(- Ca The recurrence relation is given by: +2=(- arctan(x) +c rẻ 1+n 1-n n -) c 1+n n=0,1,2,... n=0,1,2,...