Series solution Definition and 102 Threads

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...

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) =...

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}...

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...

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...

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...

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...

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}...

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...

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*}...

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 =...

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...

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...

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...

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...

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 -...

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) =...

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...

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...

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...

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...

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 =...

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...

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...

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!}...)...

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{}=...

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' =...

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...

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...

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...

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...

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...

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...

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) =...

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...

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?

[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...

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...

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.

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...

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

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

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...

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 =...

how do you solve xy`-3y=k(constant) using the power series method?

Ok I'm giving these another go. I found the following DE from a reduction of order problem and figured that it would be an alright question if I turned it into one requiring a series solution. However I'm stuck. I think it's just a matter of index shifts to get an appropriate recurrence relation...

Im given y"-xy'-y=0 at x0=1. The problem asks for recurrene relation, and the first four terms in each of two linearly independant solutions, and the general term in each solution. Whats thrwoing me off is the x0=1. I tried doing y= SUM an(x-1)^n, but when i differenetiate and plug in, i get...

Q1 :For example Solve the D.E : U''-2xU' +2u = 0 Do I write out the series solution or write somthing like : C1e^r1 + c2e^r2 ,for r1&r2 are the roots of the equation~ What is the different between these two answer? One more question, Q2 : Show function (arcsin x )^2 satisfies...

Here's our equation: \frac{d^2\psi}{du^2}+(\frac{\beta}{\alpha}-u^2)\psi=0 This is the SE for the simple harmonic oscillator. My text goes through an elaborate solution to this DE and ends up resorting to a power series solution, not for psi, but for H, where \psi=H(u)e^{-u^2/2}. The text...

Hi, I'm trying to solve a differential equation and I'm supposed to obtain a recursion formula for the coefficients of the power series solution of the following equation: w'' + (1/(1+z^2)) w = 0. The term 1/(1+z^2) I recognize as a geometric series and can be expressed as sum of 0 to...