What is Series solution: Definition and 110 Discussions

In mathematics, the power series method is used to seek a power series solution to certain differential equations. In general, such a solution assumes a power series with unknown coefficients, then substitutes that solution into the differential equation to find a recurrence relation for the coefficients.

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  1. Telemachus

    Nonhomogeneous Power Series Solution

    Hi. I have to solve: y''+xy'-2y=e^x Using series. So, this is what I did: y(x)=\sum_0^{\infty}a_n x^n y'(x)=\sum_1^{\infty}n a_n x^{n-1} y''(x)=\sum_2^{\infty}n(n-1) a_n x^{n-1} And e^x=\sum_0^{\infty}\frac{x^n}{n!} Then, using that m=n-2 for y'' and then replacing in the diff. eq...
  2. K

    Probing series solution of DE: irregular singular point

    Homework Statement Consider y''+\frac{\alpha}{x^{r}}y'+\frac{\beta}{x^{s}}y=0 Suppose we try find a solution of the form<br /> y=\sum_{n=0}^{\infty}a_{n}x^{n+\lambda} Show that if r=2 and s=1 then there is only one possible value of \lambda for which there is a formal solution in the form...
  3. Char. Limit

    Power Series Solution for y' = 4xy + 2 with Initial Condition y(0)=1

    Homework Statement I am trying to find the power series solution to y' = 4 x y + 2, with the initial condition of y(0)=1. Homework Equations The Attempt at a Solution Simple enough, I say, as I arrange the equation so I have 0 on one side. I get something like this: y' - 4 x y - 2 = 0 I then...
  4. S

    Find a recurrence formula for the power series solution around t = 0

    Homework Statement Find a recurrence formula for the power series solution around t = 0 for the differential equation: d^2 y/dt^2 + (t - 1) dy/dt + (2t - 3)y = 0 Homework Equations y = Σn=0 to inf (a_n * t^n) and formula to differentiate polynomials. The Attempt at a Solution I...
  5. H

    Series Solution around singular point

    Homework Statement x(2 - x)y'' - (x - 1)y' + 2y = 0 Find the general solution in terms of a polynomial and a series in powers of x - 1. Homework Equations The Attempt at a Solution Does the question basically ask for a series solution of the ODE at the regular point x = 1? Then y(x) =...
  6. H

    Series Solution of ODE: Finding Non-Zero Coefficients for y(x) Expansion

    Homework Statement (1 - x)y'' + xy' + xy = 0 Find the first 3 nonzero coefficients of the power series expansion about x = 0 if y(0) = -1 and y'(0) = 0Homework Equations The Attempt at a Solution y = \sum^{∞}_{n = 0}c_{n}x^{n} From above, I can say that y(0) = 1 = c_{0} and y'(0) = 0 = c_{1}...
  7. S

    Series solution to linear differential equations

    Homework Statement Given the ODE y''-ty'+y=0 where y(0)=1 and y'(0)=0 Assume y(t)=Ʃn=0 ( a(n) t^n ) (power series centered at 0) find the general form of the solution ( an=f(n) ) The Attempt at a Solution I used the initial conditions to determine the values a0=1 and a1=0...
  8. E

    Factorial question in a power series solution

    Hello, I've been working on solving the equation y''-2xy'+2py=0. where p is a positive integer. I've assumed y=\sum a_{n}x^{n} for n=0 to inf I'm getting two formulas for a_{n} One is for odd n, the other for even n, related to a_{0} and a_{1} However, the relation involves something that...
  9. J

    Power series solution for differential equation

    Homework Statement Solve the fluxional equation (y with a dot on top)/(x with a dot on top) = 2/x + 3 - x^2 by first replacing x by (x + 1) and then using power series techniques.Homework Equations dy/dx = 2/x + 3 - x^2 The Attempt at a Solution First, I believe the fluxional (y with a dot...
  10. S

    Solving ODE via Series Solution at x=0: Coefficients and Recurrence Relations

    Homework Statement Determine a series solution to the following ODE about x0 = 0: xy'' + y' + xy = 0 The Attempt at a Solution I'll try to keep this concise. I first divided through by x and made the usual guesses for the form of the series. Subbing those in gave...
  11. D

    Series solution up to a term, power series

    Homework Statement consider the initial value problem (1-x)y,,+xy,-2y=0 find the series solution up to the term with x6 Homework Equations (1-x)y,,+xy,-2y=0 The Attempt at a Solution assuming the answer has the form \Sigmaanxn that gives y,,=\Sigmananxn-1 and...
  12. T

    Power Series Solutions and Radius of Convergence for y'' + xy = 0

    Homework Statement Find 2 independent solutions which are power series in x of y'' + xy =0 and find the radius of convergence of each solution. The Attempt at a Solution \sum_{n=2}^{\infty} n(n-1)a_n x^{n-2} + x\sum_{n=0}^{\infty}a_n x^n = 0 \sum_{n=-1}^{\infty} (n+3)(n+2) a_{n+3}...
  13. T

    How to Find Power Series Solutions of Chebyshev's Equation?

    Homework Statement Chebyshev's Equation is (1-x^2) y^{\prime\prime} - xy^{\prime} + c^2 y =0 where c is a real constant. (a) Find 2 linearly independent power series solutions of Chebyshev's Equation at x=0: an even one and an odd one. (b) Hence, using the ratio test, find the radius...
  14. T

    What is the power series solution to this differential equation?

    Homework Statement http://img59.imageshack.us/img59/2091/diffeq.png [PLAIN][PLAIN]http://img684.imageshack.us/img684/6748/diffeqp.png The Attempt at a Solution Making the substitutions y= \sum_{n=0}^{\infty} a_n x^n and y^{\prime} = \sum_{n=0}^{\infty}na_nx^{n-1}, \begin{align*}...
  15. W

    Series Solution to Linear Equations: Finding the Expansion for the Term xy

    Homework Statement I just can't figure out this one term in the series. For a linear equation, one of the the terms is xy. So I need to find a series expansion of this starting at n = 0. Homework Equations The Attempt at a Solution Assume y = sum(anx^n) n = 0 dy/dx =...
  16. S

    Power series solution to a differential equation

    Homework Statement Using a power series solution, what is the solution to: (x^2-1)y" + 8xy' + 12y = 0 Homework Equations Normally these questions specify (about x0=0) but this one doesn't specify about which point. So if I use the power series equation, what am I supposed to plug in...
  17. C

    What is power series solution of y''-xy

    Homework Statement [PLAIN]http://img196.imageshack.us/img196/5241/recurrenceq.gif Homework Equations The Attempt at a SolutionThis is my attempted solution: 1) i got a recurrence relation (n+2)(n+1)a_(n+2)=a_(n-1) 2) i also used the matching coefficients method to get a2=0, a5=0, a7=0, but...
  18. T

    Infinite Series Solution for Simplifying f(m): Tips and Approximations

    I would like to simplify this series as much as possible f(m)=\sum_{n=0}^{\infty}\frac{m^n (2n)!}{(n!)^3} Approximates would also be fine. One can easily notice that (2n!) / (n!)^2 > 2^n hence I figured out that f(m) > \sum_{n=0}^{\infty}\frac{(2m)^n}{n!}=\exp(2m) but this is not the best...
  19. D

    ( ) Power series solution for ODE

    (URGENT) Power series solution for ODE Homework Statement Supose there is an infinite series solution\sum b_{n}x^{n} for u''+4(x-(1/4))^2*u+C(x) = 0 where C(x) is a function (I get it in another problem, I'll put it in the relevant equations area), determinate the coefitients b_{0} b_{1}...
  20. M

    Solving Series ODE: Finding x(0) w/ Problem Statement

    Homework Statement The Attempt at a Solution I did the "show that" part. But what is throwing me off is the x(0)=0 part. What is "x" a function of? Using the series in the square brackets, I found that when n=0, a_1 = a_0 ^2 n=1, a_2 = (a_1 * a_0)/2 n=2, a_3 = (a_0*a_2 + a_1^2 + a_2*a_0)/3...
  21. Saladsamurai

    Power Series Solution for x^2y'' - y = 0 Expanding about xo = 2: Next Steps?

    Homework Statement Solve x^2y'' - y = 0 using Power Series Solution expanding about xo = 2. The Attempt at a Solution First I expand the coefficient of y" (i.e. x2) about xo: TS[x^2]|_{x_o=2} = 4+ 4(x - 2) + (x - 2)^2 Assuming the solution takes the form: y(x) = \sum_0^{\infty}a_n(x -...
  22. Saladsamurai

    Power Series Following an example problem

    Homework Statement I am following along in an example problem and I am getting hung up on a step. We are seeking a power series solution of the DE: (x - 1)y'' + y' +2(x - 1)y = 0 \qquad(1) With the initial values y(4) = 5 \text{ and }y'(4) = 0. We seek the solution in the form y(x) =...
  23. TheFerruccio

    Another series solution ODE problem

    Homework Statement Find a basis of solutions. Homework Equations (1-x^2)y''+(1-x)y'-3y = 0 The Attempt at a Solution Using the series approach, having: y=\sum_{n=0}^{\infty}a_nx^n I ended up with an equation representing the coefficients for x^0 2a_2+a_1-3a_0 = 0 I'm...
  24. B

    Solving ODE Near x=0: Series Solution

    Homework Statement Obtain solution valid near x=0 Homework Equations (x2+1)y''+6xy'+6y=0 The Attempt at a Solution y"+6x/(x2+1)y'+6x/(x2+1)=0 In representing the solution in series notation, I'm not sure how deal with the rational function because I know I need to have all of the x...
  25. W

    How do I solve a cubic ODE using a series solution?

    Homework Statement Find the solution to the ODE via the power series: y = \Sigma_{i=0} a_j x^{2j + m} Homework Equations y' - y^3 = 0 The Attempt at a Solution I get \Sigma_{i=0} a_j (2j+m) x^{2j+m-1} - \Sigma_{i=0} (a_j)^3 x^{3(2j + m)} = 0 I don't know how to deal with the cubic...
  26. P

    How to check Fourier series solution (complex)

    Homework Statement Find the complex Fourier series for: f(t)=t(1-t), 0<t<1 Homework Equations f(t)=\sum_{n=-\infty}^{\infty}c_n{e^{i\omega_n{t}}} c_n=\frac{1}{\tau}\int_{t_0}^{t_0+\tau}e^{-i\omega_n{t}}f(t)dt \omega_n=2\pi{n}\quad\tau=1 The Attempt at a Solution I solved...
  27. A

    ODE Series Solution Near Regular Singular Point, x^2*y term?

    ODE Series Solution Near Regular Singular Point, x^2*y term? (fixed post body) Homework Statement Find the series solution (x > 0) corresponding to the larger root of the indicial equation. 5x^{2}y'' + 4xy' + 10x^{2}y = 0 Homework Equations Solution form: y =...
  28. N

    Non homogeneous differential equation - power series solution

    I am having trouble getting to a solution for this differential equation 2(x^2+2x)y' - y(x+1) = x^2+2x -------- 1 for a series solution, we have to assume y = \sum a_{n}x^n ---------- 2 if we divide equation 1 by x^2 + 2x , we get (x+1)/(x^2+2x) for the y term, which is where my problem...
  29. E

    Ordinary Differential Equation Series Solution

    Homework Statement y' = \sqrt{(1-y^2) } Initial condition y(0) = 0 a) Show y = sinx is a solution of the initial value problem. b) Look for a solution of the initial value problem in the form of a power series about x = 0. Find coefficients up to the term in x^3 in this series. Homework...
  30. J

    Series solution for ode by undetermined coefficients

    Homework Statement Obtain the Taylor series solution up to and including order 3 of the following non linear ode y'=x^2+\sin y,y(0)=\frac{\pi}{2} Homework Equations After substituting the power series form of sin(y) I get: y'=x^2+(y-\frac{y^3}{3!}+\frac{y^5}{5!}-\frac{y^7}{7!}...)...
  31. GRB 080319B

    Series Solution to ODE IVP: How to Find Initial Values?

    Homework Statement Find the series solution to the initial value problem. xy\acute{}\acute{} + y\acute{} + 2y = 0 y(1) = 2 y\acute{}(1) = 4 Homework Equations y=\sum^{\infty}_{n=0}c_{n}(x-1)^{n} t = (x-1), x = (t+1) y = \sum^{\infty}_{n=0}c_{n}t^{n} y\acute{}=...
  32. S

    Series solution of first order ODE

    Homework Statement Find two non-zero terms of the power series solution of y' = 1 + y^2 ,y(0) = 0 by using series substitution y(x) = sum (k=0 to inf) [a][/k] *x^k Homework Equations The Attempt at a Solution First take the derivative of the power series to get y' =...
  33. B

    Convergence of Series Solution

    To solve the 2nd order ode: (3x^4+4x^2+1)y'' + (6x^3-2x)y' -(6x^2-2)y=0 I used a Taylor series expansion around x=0, and I got the general solution: y=a_0(1-x^2+x^4-x^6+...)+a_1x from the recurrence relation...
  34. M

    Finding the Sum of an Alternating Series

    Homework Statement Evaluate the sum 2009^{2} - 2008^{2} + 2007^{2} - 2006^{2} + ... + 3^{2} - 2^{2} + 1^{2} Homework Equations I think that the equivalent series representation of this sum is: \sum^{2009}_{n=1}n^{2}(-1)^{n+1} The Attempt at a Solution I vaguely remember in one...
  35. D

    Series solution to 2nd order differential equation

    when using the power series to solve an ODE, is it always necessary to shift the index to 2 and 1 when taking the second and first derivatives of the power series respectively? i noticed that if i don't shift the index at all and leave them at n=0, it still works out fine? also, how...
  36. T

    Griffiths' QM book: series solution to harmonic oscillator

    I'm trying to read through Griffiths' QM book, and right now I'm on the series solution to the harmonic oscillator (ch 2). I'm having a hard time following the math (especially after equation 2.81) in this section, so if anyone has read this book, please help. My first question is about the...
  37. S

    Differential Eq- Power Series Solution

    Find a power series sol'n: (x2-1)y'' + 3xy' + xy = 0 Homework Equations let y = \Sigma (from \infty to n=0) Cnxn let y' = \Sigma (from \infty to n=1) nCnxn-1 let y'' = \Sigma (from \infty to n=2) n(n-1)Cnxn-2 The Attempt at a Solution I wrote the differential eq as...
  38. Q

    Non-Homogenous ODE Using Series Solution Method

    Homework Statement I am studying some issues on the differential equation issues. Currently I am involved with a non-homogeneous Chebyshev differential equation. The problem is to find power series solution to equation below: (1-x^2)*y’’-x*y’+y=1/(1-x) in which the x will be in...
  39. T

    Series solution to Second-order ODE

    Homework Statement Find the first four non-vanishing terms in a series solution of the form \sum from 0 to infinity of akxk for the initial value problem, 4xy''(x) + 6y'(x) + y(x) = 0, y(0) = 1 and y'(0) = -1/6 Homework Equations The Attempt at a Solution Taking the second...
  40. T

    MacLaurin Series solution to initial value problem

    Homework Statement Find the first six nonvanishing terms in the Maclaurin series solution of the initial value problem (x^2 - 3)y''(x) + 2xy'(x) = 0 where y(0) = y0 and y'(0) = y1. Homework Equations The Attempt at a Solution Should with just something like Φ(x) such that Φ(x) =...
  41. V

    Power series solution to a second order o.d.e.

    Homework Statement Find the terms up to x^5 in the power series solution of the following equation y''=(1+x^{2})y Homework Equations Power series, sum from 0 to infinity y=\sum a_{n}x^{n} The Attempt at a Solution At first I just differentiated each term separately and...
  42. S

    Infinite series solution for NON-linear ODEs?

    infinite series solution for NON-linear ODEs? Is it possible to use the infinite series method (Frobenius) to obtain general solutions of non-linear ODE's, I want to try a second order equation. Any good references where I can see how that goes exactly?
  43. Saladsamurai

    Power Series Solution to a Diff EQ

    [SOLVED] !Power Series Solution to a Diff EQ! Homework Statement Find the first 5 term of a Power series solution of y'+2xy=0 (1) Missed this class, so please bear with my attempt here.The Attempt at a Solution Assuming that y takes the form y=\sum_{n=0}^{\infty}c_nx^n...
  44. S

    Nonhomogeneous Power Series Solution

    For the fun of it, my DE book threw in a couple of problems involving nonhomogenous second order DE's in the section I'm currently going through. Although I have solved for the complementary solution, any suggestions on how to find the particular solution? For example, the one I'm looking at...
  45. S

    Power Series Solution of y''+(x^2)y=0: Is it Possible?

    y''+(x^2)y = 0 I tried to solve this problem using Power Series.But i can't make the solution in the form of series that have only two constants(a0,a1)that is, there are a0,a1, a2, a3. So i just wonder how can i make it has two constants.
  46. J

    Series Solution Coefficients for y'' - (sinx)y = cosx with Initial Conditions

    Homework Statement Find the indicated coefficients of the power series solution about x = 0 of the differential equation: y'' - (sinx)y = cosx, y(0) = -5, y'(0) = 3. y = _ + _x + _x^2 + _x^3 + _x^4 + O(x^5) Homework Equations The Attempt at a Solution This is going to be a tad...
  47. A

    Solve Series Equation: xy+(sinx)y'+2xy=3x^2, y(0)=y'(0)=1

    Hi everyone, Can anyone help me find the solution of this equation using series? xy"+(sinx)y'+2xy=3x^2 y(0)=y'(0)=1 Thanks in advance
  48. H

    Series solution near an ordinary point

    some help with series solutions I'm needing help on series solutions. It's been a while since I worked on them. Find \phi''(x_{0}) \phi'''(x_{0}) \phi''''(x_{0}) y"+xy'+y=0; y(0)=1. y'(0)=0
  49. K

    Series Solution to ODEs: Solving y''-6xy'+(6x^2-2)y=0 using Power Series

    y''-6xy'+(6x^2-2)y=0 y_{1} = _____________ I have to solve the above equation using power series.. but I am stuck. What I have so far is: y=\sum_{m=0}^\infty a_{m}x^{m} y'=\sum_{m=1}^\infty ma_{m}x^{m-1} y''=\sum_{m=2}^\infty m(m-1)a_{m}x^{m-2} = \sum_{m=0}^\infty...
  50. E

    Power Series Solution to Hydrogen Wave Function Differential Equation

    I solved the differential equation for theta portion of the hydrogen wave function using a power series solution. I got a sub n+2 = a sub n ((n(n+1)-C)/(n+2)(n+1)). I then truncated the power series at n = l to get C= l(l+1). I know need to use the recursion formula I found to find the l =...
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