# Tagged Questions

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### 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: $y=x^{c}\{a_{0}+a_{1}x+a_{2}x^{2}+.....+a_{r}x^{r}+.....\}$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ... 1answer 42 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 ... 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 ... 0answers 29 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 ... 0answers 51 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 ... 1answer 124 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 1answer 85 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! 1answer 48 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 $,then I tried to solve this but not getting the answer. if someone solve ... 3answers 50 views ### Problem about ODE and power series For each$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 ... 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 ... 0answers 78 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. ... 3answers 129 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 $$corresponding to the larger root ... 1answer 74 views ### Find one series solution for xy'' - y = 0 I have found the recurrence relation to be a_{n+1} = \frac{(a_n)}{(n+1)(n)} . I am stuck at this part because no matter what I set the initial a to be, the following term will have a problem due to ... 2answers 79 views ### Solving differential equation by using power series. Find, using the power series:$$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$$What's the convergence radius of the constructed ... 1answer 56 views ### Why don't power series methods work for linear ODE's with singularities? My math class tells me power series methods don't work for equations of the form$$f'' +p(x)f' +q(x)f = 0$$if the functions p(x) or q(x) have singularities at the point about which you're ... 1answer 98 views ### Prove \sum_{m \geq 1} {\frac{(2m-2)!}{(1-\rho)\cdots(m-\rho)} \frac{t^m}{(1-x)^{2m-1}}} is divergent How do I show that the following power series is divergent?$$ u(t,x) = \sum_{m \geq 1} {\dfrac{(2m-2)!}{(1-\rho)\cdots(m-\rho)} \dfrac{t^m}{(1-x)^{2m-1}}} $$where t is complex 1-dimensional, x ... 2answers 46 views ### a differential equation equation related to fourier series I am really struggling with this one. Any help is welcome! For equation f''(z) + p(z) f'(z) + q(z) f(z) = 0, where p(z) and q(z) are fixed polynomials. Given f(0)=f_0, f'(0)=f_1, prove that ... 2answers 62 views ### Series Solution of Second Order Linear Equation, IVP Consider the initial value problem$$y' = \sqrt{1-y^2}y(0) = 0$$Look for a solution of the IVP in the form of power series about x=0. I have started with assuming that y = \sum_{n = ... 1answer 155 views ### Finding a solution in powers of (x^2-1)y''+4xy'+2y=0 I'm working on finding the general solution of (x^2-1)y''+4xy'+2y=0 in powers. I assume the form:$$ y(x)=\sum_{n=0}^\infty C_nx^n$$My basic strategy is to first figure out each piece ... 1answer 83 views ### problem with recurrence relation for series solution for ODE I have$$y''-xy'-y=0$$and I'm trying to find the series solution around the ordinary point x_0=1. My last post I muscled through to the solution when the ordinary point was x_0=0, but this is ... 1answer 88 views ### Choice of the First Term in Legendre Polynomials The two solutions of the Legendre's Differential Equation obtained by series solution method are : and Now according to my textbook, for the useful polynomial for n equal to a positive integer, ... 2answers 103 views ### Values of k for non-trivial solutions of the differential equation y''-\left(\frac{1}{4}+\frac{k}{x}\right)y=0 where x is non-negative I attempted a power series solution of this equation in order to find the values of k that have a non-trivial solution: y''-\left(\dfrac{1}{4}+\dfrac{k}{x}\right)y=0 I am having trouble ... 3answers 133 views ### Find a power series solution centered at 0 (Differential equations Here's the problem:$$(x-1)y''+y'=0$$This is the work that I've already done:$$y=\sum_{n=0}^{\infty}a_{n}x^ny'=\sum_{n=0}^{\infty}(a_{n+1})(n+1)x^n$$... 0answers 36 views ### Power Series for Original Differential Equation The question: y"+x^2y'+2xy=0 I continue to get the incorrect answer and not sure why. I changed my indices around to make x^n all throughout and that's where the trouble starts. My answer ... 0answers 143 views ### Second Series solution y(2) for Frobenius Method I am currently solving the Frobenius Method for the question xy'' +y = 0 given the ICs y(0) = 0, y'(0)=1 I have done some work into solving that the first series solution for y_1 = ... 1answer 61 views ### power series method to solve ODE Using power series method, solve Airy’s equation$$y′′+ xy = 0$$. How do I start solving this? Thanks in advance! 1answer 119 views ### Differential Equations: Find the first four terms in each of two solutions y1 and y2 … The differential equation is y'' - xy' - y = 0 with x_0 = 1 Now, I know how to find the recurrence relation... and it's given by: a_(n+2) = [(a_(n+1) + a_(n)) / (n+2)] But I can't quite ... 0answers 27 views ### Compute coefficients of a rational expansion I am approximating certain solution of an ODE by power expansions. As it is customary, I propose an ansatz and then I check for the coefficients to satisfy the ODE. At some point of my computations I ... 1answer 162 views ### Simple Frobenius problem without recurrence relation? I am just learning frobenius method in my 'math methods in physics' class. The first problem i am trying to solve is$$ x^2y''-xy'+n^2y=0$$(where n is a constant). I know that i have to plug in the ... 0answers 29 views ### Extending a power series? I am studying a differential equation$$ 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 ... 1answer 237 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\\ ... 1answer 115 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 ... 1answer 119 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 ... 2answers 158 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: ... 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) ... 1answer 53 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} ... 1answer 121 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 ... 1answer 151 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 ... 3answers 161 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)$have power series representations of form ... 0answers 86 views ###$\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 = ...
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 ...