0
votes
1answer
16 views

Finding the coefficients of the series solution of an initial value problem

I shell present 2 questions I came across today related to this subject. I need some explanation about the meaning of the IVP I am given. We look for a solution of the form $y=\sum a_nx^n$. So after ...
2
votes
0answers
21 views

Find a series solution to $(x^2-2)y''+6xy'+4y=0$.

Find a series solution to $(x^2-2)y''+6xy'+4y=0$. A. Find the recurrence relation to $a_n$: My answer is $a_{n+2}=a_n\cdot \frac{n+4}{2(n+2)}$ which is correct. B. Using A, write two independent ...
-1
votes
3answers
40 views

Series solution to a 2nd order linear ODE

Consider the ODE $$(1+x^2)y''+3xy'+y=0$$ Find a solution in the form of $y=\sum a_nx^n$. So after doing the algebra using these 2: $$y'=\sum_{n=1}^\infty n a_n(x-x_0)^{n-1}$$ $$y''=\sum_{n=2}^\infty ...
1
vote
1answer
33 views

Series Solutions Near an Ordinary Point

I am attempting to solve this problem for practice: $y"-(x-3)y' - y = 0$ at $x_{0} = 3$. But it appears as though I don't have an idea of the best approach to employ to go about solving it. Can ...
1
vote
1answer
33 views

Can I solve an Euler differential equation by using the Frobenius method?

I'm having some trouble by trying to solve Euler equations by using the Frobenius method. For example, I'm asked to solve the Euler differential equation $$ x^2y'' + xy' - y = 0 $$ using a power ...
0
votes
0answers
11 views

Series Solution of Linear Second Equations - Difficulty Formatting Final Answer

I have been working the following differential equation: \begin{align} (1-x^2)y'' -8xy' -12 y = 0 \end{align} which has solution \begin{align} y= a_{0}\sum_{m=0}^{\infty}(m+1)(2m+1)x^{2m} + ...
1
vote
1answer
20 views

Index of Summation Shift? Power Series and Differential Equations

I have never had to index shift a summation series before, and it seems relatively straightforward, however, I am looking at an example in my textbook that doesn't make sense. I am wondering if ...
6
votes
3answers
133 views

Use Taylor Series method to solve $y''-2xy+y=0$

I am doing some practice problems for solving second order ODEs, and I am a bit stuck on this one. Here is what I have: $y''-2xy'+y=0$ Let $y = \sum_{n=0}^{\infty} C_nx^n \implies y' = ...
3
votes
1answer
58 views

Showing $y_1$ or $y_2$ are not polynomials

proof that $y_1$ or $y_2$ are not a polynomial for any $n$ $$ y_1(x)=1-\frac{n(n+1)}{2!}x^2+\frac{(n-2)n(n+1)(n+3)}{4!}x^4-+\cdots$$ $$ ...
2
votes
1answer
42 views

Trying to find 2nd power series solution

For the equation $ xy'' + 2xy' + 6e^xy = 0 $, I need to find the first 3 nonzero terms in each of two linearly independent solutions about x=0. I changed this to the form of ...
0
votes
2answers
73 views

Solution of the Legendre's ODE using Frobenius Method

This is the Legendre's differential equation given in my book: $(1-x)^{2}\ddot{y}-2x\dot{y}+k(k+1)y=0$ I solved this equation by taking: $y=x^{c}\{a_{0}+a_{1}x+a_{2}x^{2}+.....+a_{r}x^{r}+.....\}$ ...
1
vote
0answers
33 views

Power series solution for a DE with Frobenius method

The given DE is $(x²-3)y"+2xy'=0$ Since there is a singular point ($x=\pm\sqrt{3}$) I used the Frobenius method. I found two indicial relationships: $-3r(r+1)=0$ and $-3(r+1)(r+2)=0$ because I have ...
2
votes
0answers
43 views

Power series to solve differential equations?

We can use the formula $$F(x)=e^{λx} [ ρ-λμ-\dfrac{1}{2} λ^2 σ^2 ]^{-1}. (1) $$ to derive an expression for F(x) when f(x) is any integer power $x^n$. Begin by observing that for the ...
0
votes
1answer
15 views

Why not shift the index of the derivative in Euler series?

I'm reading over solving linear differential equations with analytic coefficients, and finding the solutions that are near regular singular points. In the earlier section on solving similar equations ...
0
votes
0answers
36 views

Radius of Convergence and the Frobenius Method

Consider the equation $$4xy'' + 2y'+ y = 0$$ I know that $x=4$ is a regular singular point, and in the notation that my uni uses, we say that: $$(x-x_0)^2 y'' + (x-x_0)p(x)y' + q(x)y = 0$$ where ...
0
votes
1answer
15 views

Limit when $y>>a$ of a derived solution

I am able to do part d), however I am very stuck on part e). If $y >> a$ then surely we get $\phi(x,y)$ $= \frac{1}{\pi} \Big[ tan^{-1} \Big( \frac{x+a}{y}\Big)-tan^{-1} ...
1
vote
0answers
45 views

A theoretical question regarding Frobenius method

The following is a theoretical question regarding Frobenius method. Let $b(x),c(x)$ be real functions analytic at 0. Let $b(x)=\sum_{i=0}^\infty b_ix^i, c(x)=\sum_{i=0}^\infty c_ix^i$ on $(-R,R)$. ...
1
vote
1answer
41 views

Power Series Solution to Differential Equation

The equation is $$y'' - xy' + y = 0$$ So far I have the recurrence relation - $$a_{n+2} = \dfrac{(n-1)a_n}{(n+1)(n+2)} $$ From this - $a_2 = \dfrac{-a_0}{2!}$ $a_3 = 0$ $a_4 = ...
0
votes
1answer
22 views

Series Solution To Differential Equations - Need help with one step

Would someone kindly explain to me what the logic is behind one of the steps here: http://tutorial.math.lamar.edu/Classes/DE/SeriesSolutions.aspx In Example 1 - Following on from this sentence on ...
2
votes
1answer
29 views

Power series solutions of differential equations, choosing x^n or x^(n+r)?

I cannot understand which one to use when solving differential equations by using power series solutions. For example in this question: Consider the following differential equation for $\alpha \in ...
0
votes
1answer
58 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
29 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 ...
2
votes
0answers
32 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
52 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
133 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
86 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
54 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
3answers
53 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
54 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
89 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
176 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
82 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
82 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
60 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
99 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
47 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
64 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
169 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
85 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 ...
2
votes
1answer
97 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
104 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
142 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
37 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
151 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
62 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
119 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
29 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 ...
-1
votes
1answer
137 views

Solve a differential equation using the power series method

Problem By assuming a power series solution of the form $$y(x) = \sum_{m=0}^{\infty} c_mx^m , \quad c_0 \not =0 $$ Show that the equation $ 2y'+xy=x $ has general solution ...
4
votes
1answer
176 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
30 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 ...