0
votes
1answer
21 views

Differential equation by series solution method: equating coefficients to zero

I am following the solution for a problem, and I am stuck at the following equation: $$2a_2+\sum_{n=1}^\infty \left[(n+2)(n+1)a_{n+2}-a_{n-1}\right]x^n=0\tag1$$ Now, the professor equates the ...
0
votes
1answer
24 views

What is the significance of finding the series solution of a differential equation “about a point”?

I am learning the series solution method of solving differential equations, and I am curious as to what the rationale is for finding out the solution of the equation about a particular point. It seems ...
1
vote
0answers
23 views

Solution regarding Power Series and ODE's

About 4 months ago I posted Series solution to $y''-xy'-y=0$. I ran through the analysis and it appeared that I solved the ODE . The solution seemed to be ...
0
votes
0answers
42 views

How to determine undetermined coefficients of infinite series solution of ODE by seeking the solutions that vanish at infinity?

For an ODE like $f''+p(x)f'+q(x)f=r(x)$ where $p,q,r$ are analytic functions I'm trying an infinite series solution $f=\sum_{n=0}^\infty a_n x^n$. All coefficients $a_n$ can be obtained in terms of ...
0
votes
1answer
106 views

If $f(2x)=2xf'(x)$, then find $f(x)$

If $f(x)$ is Analytic functions on $R$,and such $$2xf'(x)=f(2x)$$ Find all $f(x)$ My idea: let $$f(x)=\sum_{n=0}^{\infty}a_{n}x^n$$ so I can't Thank you
1
vote
1answer
67 views

Solving differential equations using power series

I need to solve this differential equation by power series: $$y''+3xy'+(2x^{2}+6)y=0$$ Any help is great! Thanks!
0
votes
1answer
37 views

please solve this diffrential equation question on power series

In the differential equation $y'' + (x-3)y' + y=0 $ of power series at $x_0=2$ , I took $ y=\sum_{n=0}^{\infty}a_n(x-x_0)^n $ ,then I tried to solve this but not getting the answer. if someone solve ...
0
votes
1answer
26 views

How to find singular point of following diffrential equation?

Here it is the differential equation and whether it is regular or irregular. $ x^2 y’’ + (5/3x+x^2) y’ – y/3 =0 $
0
votes
3answers
44 views

Problem about ODE and power series

For each $a \in \mathbb{Z}^+$ let the following ODE $$ x'' - \dfrac{a (a+1)}{(1 +t^2)} x = 0$$ Using power series around the origin, show that the equation has a solution $p_a(t)$ which is a ...
1
vote
1answer
51 views

Solve ODE using analytic solutions

Let the following ODE: $x'' + tx' + x = 0.$ Find the general solution $x(t) = a_0 x_1(t) + a_1 x_2(t),$ with $a_0, a_1 \in \mathbb{R}$ and $x_1(t), x_2(t)$ are $t$ power series convergent for ...
1
vote
0answers
33 views

Find the indicial equation of $(x+2)^2(x-1)y''+5(x-1)y'-\pi(x+2)y = 0$

Find all singular points of each equation, and determine whether they are regular or irregular. At each regular singular point, find the indicial equation and the exponents of singularity. ...
2
votes
3answers
77 views

Find the form of a second linear independent solution when the two roots of indicial equation are different by a integer

Consider the differential equation $$x^2y''+3(x-x^2)y'-3y=0$$ $(a)$ Find the recurrence equation and first three nonzero terms of the series solution in powers of $$ corresponding to the larger root ...
2
votes
1answer
44 views

Find one series solution for $xy'' - y = 0$

I have found the recurrence relation to be $a_{n+1} = \frac{(a_n)}{(n+1)(n)}$ . I am stuck at this part because no matter what I set the initial a to be, the following term will have a problem due to ...
0
votes
2answers
61 views

Solving differential equation by using power series.

Find, using the power series: $$y(x)=\sum_{k=0}^\infty a_{k}x^k$$ a solution for the following differential equation: $$y'(x) = -x^2y(x),\,\, y(0)=1$$ What's the convergence radius of the constructed ...
2
votes
1answer
51 views

Why don't power series methods work for linear ODE's with singularities?

My math class tells me power series methods don't work for equations of the form $$f'' +p(x)f' +q(x)f = 0$$ if the functions $p(x)$ or $q(x)$ have singularities at the point about which you're ...
1
vote
1answer
96 views

Prove $\sum_{m \geq 1} {\frac{(2m-2)!}{(1-\rho)\cdots(m-\rho)} \frac{t^m}{(1-x)^{2m-1}}} $is divergent

How do I show that the following power series is divergent? $$ u(t,x) = \sum_{m \geq 1} {\dfrac{(2m-2)!}{(1-\rho)\cdots(m-\rho)} \dfrac{t^m}{(1-x)^{2m-1}}} $$ where $t$ is complex 1-dimensional, $x$ ...
0
votes
2answers
46 views

a differential equation equation related to fourier series

I am really struggling with this one. Any help is welcome! For equation $f''(z) + p(z) f'(z) + q(z) f(z) = 0$, where $p(z)$ and $q(z)$ are fixed polynomials. Given $f(0)=f_0$, $f'(0)=f_1$, prove that ...
1
vote
2answers
52 views

Series Solution of Second Order Linear Equation, IVP

Consider the initial value problem $$y' = \sqrt{1-y^2}$$ $$y(0) = 0$$ Look for a solution of the IVP in the form of power series about x=0. I have started with assuming that $ y = \sum_{n = ...
0
votes
1answer
99 views

Finding a solution in powers of $(x^2-1)y''+4xy'+2y=0$

I'm working on finding the general solution of $(x^2-1)y''+4xy'+2y=0$ in powers. I assume the form: $$ y(x)=\sum_{n=0}^\infty C_nx^n$$ My basic strategy is to first figure out each piece ...
1
vote
1answer
65 views

problem with recurrence relation for series solution for ODE

I have $$y''-xy'-y=0$$ and I'm trying to find the series solution around the ordinary point $x_0=1$. My last post I muscled through to the solution when the ordinary point was $x_0=0$, but this is ...
1
vote
0answers
38 views

Choice of the First Term in Legendre Polynomials

The two solutions of the Legendre's Differential Equation obtained by series solution method are : and Now according to my textbook, for the useful polynomial for n equal to a positive integer, ...
5
votes
2answers
85 views

Values of $k$ for non-trivial solutions of the differential equation $y''-\left(\frac{1}{4}+\frac{k}{x}\right)y=0$ where $x$ is non-negative

I attempted a power series solution of this equation in order to find the values of k that have a non-trivial solution: $y''-\left(\dfrac{1}{4}+\dfrac{k}{x}\right)y=0$ I am having trouble ...
1
vote
3answers
110 views

Find a power series solution centered at 0 (Differential equations

Here's the problem: $$(x-1)y''+y'=0$$ This is the work that I've already done: $$y=\sum_{n=0}^{\infty}a_{n}x^n$$ $$y'=\sum_{n=0}^{\infty}(a_{n+1})(n+1)x^n$$ ...
2
votes
0answers
34 views

Power Series for Original Differential Equation

The question: $y"+x^2y'+2xy=0$ I continue to get the incorrect answer and not sure why. I changed my indices around to make x^n all throughout and that's where the trouble starts. My answer ...
1
vote
0answers
124 views

Second Series solution y(2) for Frobenius Method

I am currently solving the Frobenius Method for the question $xy'' +y = 0$ given the ICs $y(0) = 0, y'(0)=1$ I have done some work into solving that the first series solution for $y_1 = ...
2
votes
1answer
57 views

power series method to solve ODE

Using power series method, solve Airy’s equation $$y′′+ xy = 0$$. How do I start solving this? Thanks in advance!
0
votes
1answer
104 views

Differential Equations: Find the first four terms in each of two solutions y1 and y2 …

The differential equation is $y'' - xy' - y = 0$ with $x_0 = 1$ Now, I know how to find the recurrence relation... and it's given by: $a_(n+2) = [(a_(n+1) + a_(n)) / (n+2)]$ But I can't quite ...
0
votes
0answers
26 views

Compute coefficients of a rational expansion

I am approximating certain solution of an ODE by power expansions. As it is customary, I propose an ansatz and then I check for the coefficients to satisfy the ODE. At some point of my computations I ...
4
votes
1answer
141 views

Simple Frobenius problem without recurrence relation?

I am just learning frobenius method in my 'math methods in physics' class. The first problem i am trying to solve is $$ x^2y''-xy'+n^2y=0$$ (where n is a constant). I know that i have to plug in the ...
0
votes
0answers
27 views

Extending a power series?

I am studying a differential equation $$ y'(x)=g(x,y), $$ which has no analytic solution, however I have found that $y(x)$ is asymptotic to a series $$ f(x)=\sum_{k=0}^\infty a_kx^{-k} $$ as ...
3
votes
1answer
232 views

How to identify this power series as $k\sin(k/x)$?

In this question, a functional equation is solved for functions with a power series. We find a recursive formula: (copied from the answer by user achille hui) \begin{align} ( 2^1 - 3 ) a_2 &= 0\\ ...
0
votes
1answer
101 views

Solve $2(x+1)y' = y$ using Power Series.

Given the ODE: $2(x+1)y' = y$ How can I solve that using Power Series? I started to think about it: $ \\2(x+1)\sum_{n=1}^{\infty}{nc_nx^{n-1}}-\sum_{n=0}^{\infty}{c_nx^n}=0 ...
0
votes
1answer
100 views

If $f(x,y,t):= u(r) \cos ( \omega t)$, use the multivariable chain rule to obtain an ODE for $u$ from the PDE for $f$.

Let $f(x,y,t) :=u(r)\cos \omega t$, where $r= \sqrt{x^2 +y^2}$. Physics tells us the following: For $f(x,y,t)$ to describe a vibrating membrane, with $f(x,y,t)$ telling how high the mem- brane is ...
0
votes
2answers
149 views

Finding Coefficients of Power Series Expansion

Find the coefficients of $ a_{n}$ and $b_{n}$ for $ 0 ≤ n ≤ 4$ for the power series expansion of two linearly independent solutions of the ODE: $y'' -(e^{x}-1)y=0$. This is what I've tried so far: ...
1
vote
0answers
60 views

Solving ODE with negative expansion power series [duplicate]

I am solving a series of ODE, such that each DE is equal to some degree of term that I'm expanding to. For instance, one DE is this: $\xi^r\partial_r g_{rr}+2g_{tt}\partial_t\xi^t=\mathcal{O}(r)$ ...
0
votes
1answer
52 views

Simple differentiation question that I am unsure about

I am in the process of re-learning differentiation and am stuck on this as part of a larger problem. Can you explain to me why when differentiated 4 times this: $$y = \sum_{n=0}^{+\infty} ...
1
vote
1answer
114 views

Finding coefficients of a differential equation represented by power series

I am studying for a discrete mathematics exam and have gotten stuck on this question: Any function y of a real variable x that solves the diff erential equation: $$\frac{d^4y}{dx^4} -16y =0$$ may ...
0
votes
1answer
134 views

Why use series solution rather than variation of parameters?

When should we use series solution to solve a general 2nd order ODE rather than the variation of parameters? Could both methods be used to solve any 2nd order ODE or are there restrictions on when ...
2
votes
3answers
156 views

How maths can help to compute convergence?

$r'(\theta)^2 + r(\theta)^2 = \theta^2,\quad r(t=0)=0\tag{1}$ There is an interesting approach to prove that the solutions of the equation $(1)$ have power series representations of form ...
1
vote
0answers
81 views

$\sum_{k=0}^{\infty}\frac1{4^k(2k+1)}\binom{2k}{k}x^{2k+1}\sum_{k=0}^{\infty}(-1)^k\frac1{4^k}\binom{2k}{k}x^k =$

Prove that for $|x|<1$, $\sum_{k=0}^{\infty}\frac1{4^k(2k+1)}\binom{2k}{k}x^{2k+1}\sum_{k=0}^{\infty}(-1)^k\frac1{4^k}\binom{2k}{k}(-x^2)^k = ...
5
votes
1answer
410 views

When solving an ODE using power series method, Why do we need to expand the solution around the singular point?

When solving a differential equation using series expansion method, if it has the following form : $$y''+\frac{p(x)}{x}y'+\frac{q(x)}{x^2}y=0$$ ; where $p$ and $q$ are analytic at $x_0$; if we want ...
0
votes
1answer
350 views

Method for solving ODE with power series

when trying to solve second order linear homogeneous variable coefficient ODEs using a power series method, there seem to be two different general forms cropping up in my notes. The first uses an ...
0
votes
1answer
131 views

Find the first 5 terms of the expansion in a power series

Find the first 5 terms of the expansion in a power series $$y′=xe^{x}+2y^{2}$$ I've got a riccati equation $$ x e^{x}+2y^{2}, y(0)=0$$ After solving: $$y=e^{x}(x-1)+\frac{2}{3}y^{3} - 1$$ And I ...
2
votes
1answer
134 views

Hermite's equation of order $\alpha$

Show that the general solution of Hermite's equation of order $\alpha$: $${y}''-2x{y}'+2\alpha y=0$$ $$is$$ $$y(x)=c_{0}y_{1}(x)+c_{1}y_{2}(x)$$ where $y_{1}(x)$ and $y_{2}(x)$ are power series ...
0
votes
1answer
175 views

Solving the second order differential equation: y' =( x^2)*y as a power series

What is the proper way to do this problem as a power series? The way I'm doing it, I end up with a very complicated term. how I'm doing it: take the series for y and assume it's of the form ...
1
vote
0answers
95 views

Relating terms in differential equation with power series

Having problems with a task on a differential equation containing a power series. Given $$\frac{dx}{dt} = \lambda x + \sum_{n=2}^\infty b_n x^n$$ $$\frac{dy}{dt} = \lambda y$$ $$x(y) = y + ...
1
vote
3answers
310 views

Index of summation shift

I'm learning about power series in Differential Equations. Right now I'm learning about shifting summations and something that is bothering me is the following: Take the equation $$F(x) = (x-3)y' + ...
1
vote
1answer
183 views

using power series expansion to find a holomorphic function which solves a differential equation

Using power series expansions, find a function $f$ which is holomorphic on the unit disk $D:=$ {$z\in\mathbb C:|z|<1$} and solves the differential equation $(1-z^2)f''(z)-4zf'(z)-2f(z)=0$ for ...
2
votes
2answers
56 views

Running into trouble with this differential equation

We're having trouble with this differential equation: $xy'' + x^2y + y = 0$ We figured it is regular singular because there are no singular points. We assumed a frobenius solution: $y = ...
2
votes
1answer
137 views

Convergence of a power series function

Consider the following differential equation: $$w''(x)+p(x)w'(x)+q(x)w(x)=r(x)$$ with the initial condition of $w(0)=w_0,\ w'(0)=w_1$, and $$w_{n+2}=\frac{r_{n+2}-(n+1)p_0w_{n+1}-\sum_{k=0}^n w_k ...