Consider the differential equation 2x2y" + 3xy' + (2x - 1)y=0.The indicial equation is 2r²+r-1=0.The recurrence relation is ck[2(k + r) + (k+r-1) + 3(k+r)-1]+2Ck-1=0.A series solution corresponding to the indicial root r = - 1 isy=x-¹[1 + Σk=1kxk], whereOack=(-2)k/[k!(-1).1.3...(2k-3)]Obck=(-2)k/[k!(-1)(2k-3)!]Ock=(-2)/[k!1.3...(2k-5)]O d.ck = -2k/[k! 1.3...(2k-3)]Oeck=(-2)/[k!(-1).1.3...(2k-1)]

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Answered: Consider the differential equation…

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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Can you help me with the third part?
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.
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 +...
I know the answer is x+(1/2)(x^2)+(1/2)(x^3)+(13/24)(x^4)+(11/15)(x^5) but can you help me on verifying the answer? Somehow it matches the original equation is what I'm looking for.
2- Consider the differential equation (²+3)y" - 4y' = 0. The recurrence relation for the coefficients an of the power series solution an" about the ordinary point o=0 is given by b) an+2= an+2= an+2= none of these an+2= 12-0 4nan+1 + (n² = n)an 3n² +9n+6 n22 (4n+4)an+1-(n² - n)an 3n² +9n+6 (n + 4)an+1 +n²an 3n² +9n+6 an+1 + (n²-n)an 3n² +9n+6 1 " n>2 n≥ 2 n22
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
Find two power series solutions of the given differential equation about the ordinary point x = 0. (x2+1)y" 6y = 0 Oy₁ = 1 + 3x² + x²-to 1,6 Oy₁ = 1 + 3x² + 5x4 + 7x6 + Oy₁ = 1 + 3x² + x4 + +6 5 + + and y₂ = x + x³ and Y2 = x + 2x³ + 3x5 + 4x7. and y₂ = x - x³ 34 ₁=1+²+x+6+ ... and y₂ = x+3x² Oy₁ = 1 - 3x² + 5x4 – 7x6 + and y₂ = x - 2x³ + 3x5 - 4x² + .
By integrating the recurrence relation for derivatives of the Bessel functions of the ﬁrst kind, show that
The technique works for equations of any order, for example, first order. If is a solution of the differential equation ak+2 then its coefficients a satisfy the following recurrence relation. For k ≥ 0, We also have 2 k+2 2 48 a₁ = -2 ao. help (numbers) Therefore, write the first few terms of a solution where y(0) = 2 y = 2 ak+1+ 4. 1 k+2 +-4 x² +-2 + help (numbers) x+ y=[akak k=0 2³+ y + (2x + 2)y=0, a. help (formulas)
Consider the differential equation (x2−1)y′′+[1−(a+b)]xy′+aby=0, (11.7.9) where a and b are constants.

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