# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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
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### 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!
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### 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 ...
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### How to find singular point of following diffrential equation?

Here it is the differential equation and whether it is regular or irregular. $x^2 y’’ + (5/3x+x^2) y’ – y/3 =0$
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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 = ... 1answer 411 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 ... 1answer 352 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 ... 1answer 132 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 ... 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 ... 1answer 188 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 ... 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 $$\frac{dx}{dt} = \lambda x + \sum_{n=2}^\infty b_n x^n$$ $$\frac{dy}{dt} = \lambda y$$ $$x(y) = y + ... 3answers 318 views ### Index of summation shift I'm learning about power series in Differential Equations. Right now I'm learning about shifting summations and something that is bothering me is the following: Take the equation$$F(x) = (x-3)y' + ... 1answer 187 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 ... 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 = ...
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 ...