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. dim_d00m

    A Recurrence relations for series solution of differential equation

    I am currently looking at section IIA of the following paper: https://arxiv.org/pdf/gr-qc/0511111.pdf. Eq. (2.5) proposes an ansatz to solve the spheroidal wave equation (2.1). This equation is $$ \dfrac{d}{dx} \left((1-x^2) \dfrac{d}{dx}S_{lm} \right) + \left(c^2x^2 + A_{lm} -...
  2. M

    A Inverse ODE, Green's Functions, and series solution

    Hi PF! One way to solve a simple eigenvalue problem like $$y''(x)+\lambda y(x) = 0,\\ y(0)=y(1)=0$$ (I realize the solution's amplitude can be however large, but my point here is not to focus on that) is to solve the inverse problem. If we say ##A[u(x)] \equiv d^2_x u(x)## and ##B[u(x)] \equiv...
  3. L

    Series solution of a second order ordinary DE

    Homework Statement Use the power series method to solve the initial value problem: ##(x^2 +1)y'' - 6xy' + 12y = 0, y(0) = 1, y'(0) = 1## Homework EquationsThe Attempt at a Solution The trouble here is that after the process above I end up with ##c_{k+2} = -...
  4. J

    Series Solution to Second Order DE

    Homework Statement Consider a power series solution about x0 = 0 for the differential equation y'' + xy' + 2y = 0. a) Find the recurrence relations satisfied by the coefficients an of the power series solution. b) Find the terms a2, a3, a4, a5, a6, a7, a8 of this power series in terms of the...
  5. C

    What is the Definition of Period in Fourier Series?

    Homework Statement Homework Equations The Attempt at a Solution a0=4 an=8/Pi*n Heres a written solution https://gyazo.com/57e11d1e7a360914db8aec167beb6b39.png
  6. Eclair_de_XII

    Finding a power series solution to a differential equation?

    Homework Statement "Find the recurrence relation in the power series solution for ##y''-xy'-y=0## centered about ##x_0=1##." Homework Equations ##y=\sum_{n=0}^\infty a_nx^n## Answer as given in book: ##(n+2)a_{n+2}-a_{n+1}-a_n=0## The Attempt at a Solution ##y=\sum_{n=0}^\infty a_n(x-1)^n##...
  7. JamesonS

    Solve Diff. Eq. using power series

    Homework Statement \begin{equation} (1-x)y^{"}+y = 0 \end{equation} I am here but do not understand how to combine the two summations: Mod note: Fixed LaTeX in following equation. $$(1-x)\sum_{n=0}^{\infty}(n+2)(n+1)a_{n+2}x^n+\sum_{n=0}^{\infty}a_nx^n = 0$$
  8. S

    Series solution for differential equation

    <OP warned about not using the homework template> Obtain a series solution of the differential equation x(x − 1)y" + [5x − 1]y' + 4y = 0Do I start by solving it normally then getting a series for the solution or assume y=power series differentiate then add up the series? I did the latter and...
  9. C

    I  Simplifying Summation Algebra with Differential Equations

    Hi, I'm working with series solutions of differential equations and I have come across something that has troubled me other courses as well. given that \begin{equation} \sum_{n=0}^{\infty} c_{n+2}x^n+e^{-x} \sum_{n=0}^{\infty}c_{n}x^n \\ \text{where}\\...
  10. Odious Suspect

    I Power Series solution to dy/dx=x+y

    This is from an example in Thomas's Classical Edition. The task is to find a solution to ##\frac{dy}{dx}=x+y## with the initial condition ##x=0; y=1##. He uses what he calls successive approximations. $$y_1 = 1$$ $$\frac{dy_2}{dx}=y_1+x$$ $$\frac{dy_3}{dx}=y_2+x$$ ...
  11. W

    Series solution of ODE near singular points with trig

    Homework Statement Given the differential equation (\sin x)y'' + xy' + (x - \frac{1}{2})y = 0 a) Determine all the regular singular points of the equation b) Determine the indicial equation corresponding to each regular point c) Determine the form of the two linearly independent solutions...
  12. H

    Power series solution, differential equation question

    I can not find a solid explanation on this anywhere, so forgive me if this has been addressed already. Given something like y''+y'-(x^2)y=1 or y''+2xy'-y=x, how do I approach solving a differential with a power series solution when the differential does not equal zero? Would I solve the left...
  13. P

    Series Solution of Differential Equations - Real or Fake?

    Hi guys, I was browsing in regards to differential equations, the non-linear de and came up with this site in facebook: https://www.facebook.com/nonlinearDE Are these people for real? Can just solve any DE like that, come up with a series? Not an expert in this area, so I do not know what if...
  14. D

    Fourier series solution of wave equation

    Homework Statement Suppose a horizontally stretched string is heavy enough for the effects of gravity to be significant, so that the wave equation must be replaced by ##u_{tt} = c^2u_{xx} - g## where ##g## is the acceleration due to gravity. The boundary conditions are ##u(0,t) = u(l,t) = 0##...
  15. Destroxia

    Series solution with regular singular points?

    1. Homework Statement ##x^{2}y'' + (x^{2} + 1/4)y=0## 3. The Attempt at a Solution First I found the limits of a and b, which came out to be values of a = 0, and b = 1/4 then I composed an equation to solve for the roots: ##r^{2} - r + 1/4 = 0## ##r=1/2## The roots didn't differ by an...
  16. Destroxia

    Finding series solution for the differential equation

    Homework Statement y'' - xy' + xy = 0 around x0=0 Find a solution to the 2nd order differential equation using the series solution method.Homework Equations Assume some function y(x)= ∑an(x-x0)n exists that is a solution to the above differential equation.The Attempt at a Solution How...
  17. G

    Frobenius Solution to 2xy'' +5y' -4xy = 0 at x = 0

    After determining that x = 0 is a regular singular point of this equation, the frobenius method allows you to assume that y = Σanxn + r. Then I can take the first and second derivative of this assumption and plug it into the DE and begin solving with the general method: Multiply the...
  18. Q

    Calculating Infinite Sine Sum with Ratio Test | x and t Real Numbers

    Homework Statement I have this exercise: Calculate ##\sum\limits_{k=0}^\infty t^{k}sin{(kx)}## Where x and t are real and t is between 0 and 1. Homework Equations ? The Attempt at a Solution The ratio test says that this sum does have a limit, and tk obviously converges, as t is between 0 and...
  19. P

    Power series solution to degree 2 ODE

    Homework Statement (x+1)y'' - (x-1)y' - y = 0 centred around x=1 y(1) = 2, y'(1) = 3 The Attempt at a Solution I know I am supposed to get two power series, one with a0 and one with a1 but when I am trying to figure out a pattern, I keep getting both a0 and a1 in all of my terms. So I end up...
  20. Y

    Series with binomial coefficients

    Hi all, I have an apparently simple equation. I copy here its Mathematica code: Sum[(p/(1 - p))^s*(q/(1 - q))^s*Binomial[n, s]*(Binomial[m - 1, s]*(p*q*(m + n) + (2*m - 1)*(-p - q + 1))), {s, 0, n}] == Sum[(p/(1 - p))^s*(q/(1 - q))^s*Binomial[n, s]*((-(-p - q + 1))*Binomial[m - 2, s] +...
  21. Soumalya

    TextBooks for Some Topics in Mathematics

    Hi, I need suggestions for picking up some standard textbooks for the following set of topics as given below: Ordinary and singular points of linear differential equations Series solutions of linear homogenous differential equations about ordinary and regular singular points...
  22. H

    Series Solution of Differential equations

    Working through Mathematical Methods in the Physical Sciences by Boas, on the chapter on Series Solutions of Differential Equations, Boas works the example: y' = 2xy Boas differentiates the series representation of y yielding y', substitutes both into the original equation, and expands...
  23. teroenza

    Series solution to nonlinear differential equation

    Homework Statement By truncating the differential equation below at n=12, derive the form of the solution, obtaining expressions for all the ancoefficients in terms of the parameter \lambda .Homework Equations The ODE is: \frac{\mathrm{d^2}\phi }{\mathrm{d} x^2} = \frac{\phi^{3/2}}{x^{1/2}}...
  24. N

    MHB Solving Regular Singularities with Frobenius Expansion: Series Solution Question

    Referring to the attached image. I have found the solution first solution, it had regular singularities of $x_0=1$ annd $x_0=0$ so we can use frobenius expansion. The indicial equation is r= 1 or 0, and the solution I found was for r=1. ****Is there only going to be one solution for this...
  25. V

    Power series solution to differential equation

    Homework Statement Find the power series solution of the differential equation y''-\frac{2}{(1-x)^2}y=0 around the point ##x=0##. Homework Equations y=\sum_{n=0}^\infty{}c_nx^n y'=\sum_{n=0}^\infty{}c_{n+1}(n+1)x^n y''=\sum_{n=0}^\infty{}c_{n+2}(n+2)(n+1)x^n The Attempt...
  26. B

    Power Series Solution of Differential Equation

    Homework Statement (x^2)y' = y Homework Equations The Attempt at a Solution Plugging in series everywhere I get the equation \sum na_{n}x^{n+1} = \sum a_{n}x^{n}. I try to set the coefficients for the corresponding powers equal, but when I do I don't get the correct answer. I also...
  27. vanceEE

    Power Series Solution y' = xy

    Homework Statement $$y' = xy$$Homework Equations $$y = a_{0} + a_{1}x + a_{2}x^{2}+... = \sum\limits_{n=0}^∞ a_{n}x^{n}$$ $$xy = a_{0}x + a_{1}x^{2} + a_{2}x^{3}+... = \sum\limits_{n=0}^∞ a_{n}x^{n+1}$$ $$y' = a_{1} + 2a_{2}x + 3a_{3}x^{2}... = \sum\limits_{n=1}^∞ n a_{n}x^{n-1}$$The Attempt...
  28. ChrisVer

    Series Solution to Differential Equation

    I have to solve the differential equation y''+(1-t) y' + y= sin(2t) can someone judge this? How could I continue it? y=\sum_{n=0}^{∞}{a_{n} t^{n}} y'=\sum_{n=1}^{∞}{a_{n} n t^{n-1}} y''=\sum_{n=2}^{∞}{a_{n} n(n-1) t^{n-2}} sin(2t)=\sum_{n=0}^{∞}{\frac{2^{2n}}{2n!} t^{2n}} y''+(1-t) y'...
  29. P

    Find the Fourier series solution to the differential equation

    Find the Fourier series solution to the differential equation x"+x=t It's given that x(0)=x(1)=0 So, I'm trying to find a Fourier serie to x(t) and f(t)=t, and I'm know it must a serie of sin... So here's my question...the limits of integration to the Bn, how do I define them? Will...
  30. F

    Series solution to diffeq, stuck on matching the indices

    Homework Statement find the series solution to y''+x^2*y'+y=0 Homework Equations y=summation from n=0 to infinity Cn*x^n The Attempt at a Solution y=sum from 0 to inf Cnxn x^2*y'=sum from 1 to inf nC n xn+1 = sum from 2 to inf (n-1) C n-1 xn = sum from 1 to inf (n-1) C n-1 xn...
  31. G

    Few questions about series solution of ODEs

    Consider the ODE x(x-1)y''-xy'+y=0. I need help in identifying the method of solution (power series or frobenius) for this ODE. Using the formulae \stackrel{limit}{_{x→x_{o}}}\frac{q(x)+r(x)}{p(x)} and \stackrel{limit}{_{x→x_{o}}}\frac{(x-x_{o})q(x)+(x-x_{o})^{2}r(x)}{p(x)} , where...
  32. C

    Power Series Solution to Linear ODE

    Homework Statement Let y(x)=\sumckxk (k=0 to ∞) be a power series solution of (x2-1)y''+x3y'+y=2x, y(0)=1, y'(0)=0 Note that x=0 is an ordinary point. Homework Equations y(x)=\sumckxk (k=0 to ∞) y'(x)=\sum(kckxk-1) (k=1 to ∞) y''(x)=\sum(k(k-1))ckxk-2 (k=2 to ∞) The Attempt at a Solution...
  33. R

    Finding Recursion Relations for Coefficients in Power Series Solutions for ODEs

    Homework Statement I am trying to find the recursion relation for the coefficients of the series around x=0 for the ODE: y'''+x^2y'+xy=0 The Attempt at a Solution Therefore letting: y=\sum_{m=0}^\infty y_mx^m \therefore y'=\sum_{m=1}^\infty my_mx^{m-1} \therefore...
  34. C

    Second Order Linear ODE - Power Series Solution to IVP

    Homework Statement Let y(x)=\sumckxk (k=0 to ∞) be a power series solution of (x2-1)y''+x3y'+y=2x, y(0)=1, y'(0)=0 Note that x=0 is an ordinary point. Homework Equations y(x)=\sumckxk (k=0 to ∞) y'(x)=\sum(kckxk-1) (k=1 to ∞) y''(x)=\sum(k(k-1))ckxk-2 (k=2 to ∞) The Attempt at a Solution...
  35. S

    Series solution of a differential equation

    Homework Statement Solve the following differential equation by power series and also by an elementary method. Verify that the series solution is the power series expansion of your other solution. y'' = - 4y Homework Equations The Attempt at a Solution y'' = - 4y \\ \frac{d^{2}y}{dx^{2}}...
  36. C

    Fourier Series Solution of 1-D Heat Flow

    Homework Statement Length of rod = 1 Initial Conditions: u(x,0)=sin(πx) Boundary conditions: u(0,t)=0 and u(1,t)=5. Alright I am supposed to find the temperature at all times, but I am curious about the setup of the problem itself. When x = 1, the boundary condition says...
  37. L

    Series solution to DE about ordinary point

    Homework Statement Find two power series solutions of the DE (x+2)y'' + xy' - y = 0 about the ordinary point x = 0 . Include at least first four nonzero terms for each of the solutions. 2. The attempt at a solution I distributed the y'' term and substituted y = Ʃ0inf cnxn...
  38. M

    Series Solution of an ODE: Finding a Non-Recursive Formula

    Homework Statement Solve for y' = x^2y The Attempt at a Solution There's something that's been really bothering me about this question and similar ones. We assume that the solution to the ODE will take the form y = \sum_{n=0}{a_nx^n} After finding y', plugging in the expressions...
  39. J

    Series Solution for O.D.E (Frobenius method?)

    Homework Statement (For Physics 306: Theoretical methods of Physics... Text book: Mathematical Tools for Physics (really good!)... Assumption: 300 level class = considerably junior level class) -- Find a series solution about x=0 for y''+ysec(x) = 0, at least to a few terms. (Ans...
  40. S

    MHB Power series solution for Log(1+x)

    Show that, \[\log(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}+\cdots\]
  41. T

    Simple first-order series solution

    Homework Statement The question is y' - xy = 0 I have to solve it using series solutions Homework Equations The Attempt at a Solution I use y = Ʃ from 0 to infinity a_n x^n and took the derivative. I plugged it into the equation I got the recurrence relation to be a1 = 0...
  42. D

    Missing coefficient for Series Solution for DE

    I can't for the life of me figure out where C_{0} went. I scanned my work here..I've looked back and forth through my book and other texts, it always seems like all the coefficients are accounted for and/or they equal zero. As it stands, I only have C_{1} Thanks! Full size...
  43. B

    Series solution for y +x*y=0

    Series solution for y"+x*y=0 Working on recurance realtion. Get to (sum(n=2))n*(N-1)*a(n)*X^(n-2)+(sum(n=0))a(n)*x^(n) Try several things but not sure if their correct.
  44. D

    Series Solution to PDE with Inhomogeneous Term | Step-by-Step Guide

    Homework Statement Consider the PDE which has the solution The Attempt at a Solution So what I am having trouble is solving it using this method. I am going to say that my $$u(x,t) = \sum_{n=1}^{\infty} u_n(t) \sin(nx)$$ and $$x \sin(t) =...
  45. W

    MHB Series Solution for DE: A Guide to Solving with Zill's Book

    Hey! I'm having problems with finding the general solution of this DE, using series. I have readed the Zill book, but I don't know how to start solving. Any help is appreciated! y'' - 4xy' -4y = e^x
  46. P

    Series solution about a regular singular point (x=0) of xy''-xy'-y=0

    Homework Statement Find the indicial equation and find 2 independent series solutions for the DE: xy''-xy'-y=0 about the regular singular point x=0 Homework Equations y=Ʃ(0→∞) Cnxn+r y'=Ʃ(0→∞) Cn(n+r)xn+r-1 y''=Ʃ(0→∞) Cn(n+r)(n+r-1)xn+r-2 The Attempt at a Solution Finding the...
  47. S

    Theoretical/non-tedious question about power series solution of y'' + y = 0

    1. "Homework Statement Find a recurrence formula for the power series solution around x = 0 for the differential equation given in the previous problem." The previous problem says: "Determine whether x = 0 is an ordinary point of the differential equation y'' + y = 0." Homework...
  48. O

    Series Solution Near an Ordinary Point

    Homework Statement Determine φ''(x0), φ'''(x0), and φ(4)(x0) for the given point x0 if y=φ(x) is a solution of the given initial value problem. y'' + (sinx)y' + (cosx)y = 0 y(0) = 0; y'(0) = 1 Homework Equations y = φ(x) = Ʃan(x-x0)n The Attempt at a Solution I started off by...
  49. Telemachus

    Series solution, second order diff. eq.

    Hi there. I have this differential equation: x^4y''+2x^3y'-y=0 And I have to find one solution of the form: \sum_0^{\infty}a_nx^{-n},x>0 So I have: 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-2} Then, replacing in the diff...