What is Series solution: Definition and 110 Discussions
In mathematics, the power series method is used to seek a power series solution to certain differential equations. In general, such a solution assumes a power series with unknown coefficients, then substitutes that solution into the differential equation to find a recurrence relation for the coefficients.
I am currently looking at section IIA of the following paper: https://arxiv.org/pdf/gr-qc/0511111.pdf. Eq. (2.5) proposes an ansatz to solve the spheroidal wave equation (2.1). This equation is
$$ \dfrac{d}{dx} \left((1-x^2) \dfrac{d}{dx}S_{lm} \right) + \left(c^2x^2 + A_{lm} -...
Hi PF!
One way to solve a simple eigenvalue problem like
$$y''(x)+\lambda y(x) = 0,\\
y(0)=y(1)=0$$
(I realize the solution's amplitude can be however large, but my point here is not to focus on that) is to solve the inverse problem. If we say ##A[u(x)] \equiv d^2_x u(x)## and ##B[u(x)] \equiv...
Homework Statement
Use the power series method to solve the initial value problem:
##(x^2 +1)y'' - 6xy' + 12y = 0, y(0) = 1, y'(0) = 1##
Homework EquationsThe Attempt at a Solution
The trouble here is that after the process above I end up with ##c_{k+2} = -...
Homework Statement
Consider a power series solution about x0 = 0 for the differential equation y'' + xy' + 2y = 0.
a) Find the recurrence relations satisfied by the coefficients an of the power series solution.
b) Find the terms a2, a3, a4, a5, a6, a7, a8 of this power series in terms of the...
Homework Statement
Homework Equations
The Attempt at a Solution
a0=4
an=8/Pi*n
Heres a written solution
https://gyazo.com/57e11d1e7a360914db8aec167beb6b39.png
Homework Statement
"Find the recurrence relation in the power series solution for ##y''-xy'-y=0## centered about ##x_0=1##."
Homework Equations
##y=\sum_{n=0}^\infty a_nx^n##
Answer as given in book: ##(n+2)a_{n+2}-a_{n+1}-a_n=0##
The Attempt at a Solution
##y=\sum_{n=0}^\infty a_n(x-1)^n##...
Homework Statement
\begin{equation}
(1-x)y^{"}+y = 0
\end{equation}
I am here but do not understand how to combine the two summations:
Mod note: Fixed LaTeX in following equation.
$$(1-x)\sum_{n=0}^{\infty}(n+2)(n+1)a_{n+2}x^n+\sum_{n=0}^{\infty}a_nx^n = 0$$
<OP warned about not using the homework template>
Obtain a series solution of the diﬀerential equation x(x − 1)y" + [5x − 1]y' + 4y = 0Do I start by solving it normally then getting a series for the solution or assume y=power series differentiate then add up the series?
I did the latter and...
Hi, I'm working with series solutions of differential equations and I have come across something that has troubled me other courses as well. given that
\begin{equation}
\sum_{n=0}^{\infty} c_{n+2}x^n+e^{-x} \sum_{n=0}^{\infty}c_{n}x^n \\
\text{where}\\...
This is from an example in Thomas's Classical Edition. The task is to find a solution to ##\frac{dy}{dx}=x+y## with the initial condition ##x=0; y=1##. He uses what he calls successive approximations.
$$y_1 = 1$$
$$\frac{dy_2}{dx}=y_1+x$$
$$\frac{dy_3}{dx}=y_2+x$$
...
Homework Statement
Given the differential equation
(\sin x)y'' + xy' + (x - \frac{1}{2})y = 0
a) Determine all the regular singular points of the equation
b) Determine the indicial equation corresponding to each regular point
c) Determine the form of the two linearly independent solutions...
I can not find a solid explanation on this anywhere, so forgive me if this has been addressed already.
Given something like y''+y'-(x^2)y=1 or y''+2xy'-y=x, how do I approach solving a differential with a power series solution when the differential does not equal zero?
Would I solve the left...
Hi guys,
I was browsing in regards to differential equations, the non-linear de and came up with this site in facebook:
https://www.facebook.com/nonlinearDE
Are these people for real? Can just solve any DE like that, come up with a series? Not an expert in this area, so I do not know what if...
Homework Statement
Suppose a horizontally stretched string is heavy enough for the effects of gravity to be significant, so that the wave equation must be replaced by ##u_{tt} = c^2u_{xx} - g## where ##g## is the acceleration due to gravity. The boundary conditions are ##u(0,t) = u(l,t) = 0##...
1. Homework Statement
##x^{2}y'' + (x^{2} + 1/4)y=0##
3. The Attempt at a Solution
First I found the limits of a and b, which came out to be values of a = 0, and b = 1/4
then I composed an equation to solve for the roots:
##r^{2} - r + 1/4 = 0## ##r=1/2##
The roots didn't differ by an...
Homework Statement
y'' - xy' + xy = 0 around x0=0
Find a solution to the 2nd order differential equation using the series solution method.Homework Equations
Assume some function y(x)= ∑an(x-x0)n exists that is a solution to the above differential equation.The Attempt at a Solution
How...
After determining that x = 0 is a regular singular point of this equation, the frobenius method allows you to assume that y = Σanxn + r. Then I can take the first and second derivative of this assumption and plug it into the DE and begin solving with the general method:
Multiply the...
Homework Statement
I have this exercise: Calculate
##\sum\limits_{k=0}^\infty t^{k}sin{(kx)}##
Where x and t are real and t is between 0 and 1.
Homework Equations
?
The Attempt at a Solution
The ratio test says that this sum does have a limit, and tk obviously converges, as t is between 0 and...
Homework Statement
(x+1)y'' - (x-1)y' - y = 0
centred around x=1
y(1) = 2, y'(1) = 3
The Attempt at a Solution
I know I am supposed to get two power series, one with a0 and one with a1 but when I am trying to figure out a pattern, I keep getting both a0 and a1 in all of my terms.
So I end up...
Hi,
I need suggestions for picking up some standard textbooks for the following set of topics as given below:
Ordinary and singular points of linear differential equations
Series solutions of linear homogenous differential equations about ordinary and regular singular points...
Working through Mathematical Methods in the Physical Sciences by Boas, on the chapter on Series Solutions of Differential Equations, Boas works the example:
y' = 2xy
Boas differentiates the series representation of y yielding y', substitutes both into the original equation, and expands...
Homework Statement
By truncating the differential equation below at n=12, derive the form of the solution, obtaining expressions for all the ancoefficients in terms of the parameter \lambda .Homework Equations
The ODE is:
\frac{\mathrm{d^2}\phi }{\mathrm{d} x^2} = \frac{\phi^{3/2}}{x^{1/2}}...
Referring to the attached image.
I have found the solution first solution, it had regular singularities of $x_0=1$ annd $x_0=0$ so we can use frobenius expansion.
The indicial equation is r= 1 or 0, and the solution I found was for r=1.
****Is there only going to be one solution for this...
Homework Statement
Find the power series solution of the differential equation
y''-\frac{2}{(1-x)^2}y=0
around the point ##x=0##.
Homework Equations
y=\sum_{n=0}^\infty{}c_nx^n
y'=\sum_{n=0}^\infty{}c_{n+1}(n+1)x^n
y''=\sum_{n=0}^\infty{}c_{n+2}(n+2)(n+1)x^n
The Attempt...
Homework Statement
(x^2)y' = y
Homework Equations
The Attempt at a Solution
Plugging in series everywhere I get the equation \sum na_{n}x^{n+1} = \sum a_{n}x^{n}. I try to set the coefficients for the corresponding powers equal, but when I do I don't get the correct answer. I also...
I have to solve the differential equation
y''+(1-t) y' + y= sin(2t)
can someone judge this?
How could I continue it?
y=\sum_{n=0}^{∞}{a_{n} t^{n}}
y'=\sum_{n=1}^{∞}{a_{n} n t^{n-1}}
y''=\sum_{n=2}^{∞}{a_{n} n(n-1) t^{n-2}}
sin(2t)=\sum_{n=0}^{∞}{\frac{2^{2n}}{2n!} t^{2n}} y''+(1-t) y'...
Find the Fourier series solution to the differential equation x"+x=t
It's given that x(0)=x(1)=0
So, I'm trying to find a Fourier serie to x(t) and f(t)=t, and I'm know it must a serie of sin...
So here's my question...the limits of integration to the Bn, how do I define them? Will...
Homework Statement
find the series solution to y''+x^2*y'+y=0
Homework Equations
y=summation from n=0 to infinity Cn*x^n
The Attempt at a Solution
y=sum from 0 to inf Cnxn
x^2*y'=sum from 1 to inf nC n xn+1 = sum from 2 to inf (n-1) C n-1 xn = sum from 1 to inf (n-1) C n-1 xn...
Consider the ODE x(x-1)y''-xy'+y=0.
I need help in identifying the method of solution (power series or frobenius) for this ODE.
Using the formulae \stackrel{limit}{_{x→x_{o}}}\frac{q(x)+r(x)}{p(x)} and \stackrel{limit}{_{x→x_{o}}}\frac{(x-x_{o})q(x)+(x-x_{o})^{2}r(x)}{p(x)} , where...
Homework Statement
Let y(x)=\sumckxk (k=0 to ∞) be a power series solution of
(x2-1)y''+x3y'+y=2x, y(0)=1, y'(0)=0
Note that x=0 is an ordinary point.
Homework Equations
y(x)=\sumckxk (k=0 to ∞)
y'(x)=\sum(kckxk-1) (k=1 to ∞)
y''(x)=\sum(k(k-1))ckxk-2 (k=2 to ∞)
The Attempt at a Solution...
Homework Statement
I am trying to find the recursion relation for the coefficients of the series around x=0 for the ODE: y'''+x^2y'+xy=0
The Attempt at a Solution
Therefore letting:
y=\sum_{m=0}^\infty y_mx^m
\therefore y'=\sum_{m=1}^\infty my_mx^{m-1}
\therefore...
Homework Statement
Let y(x)=\sumckxk (k=0 to ∞) be a power series solution of
(x2-1)y''+x3y'+y=2x, y(0)=1, y'(0)=0
Note that x=0 is an ordinary point.
Homework Equations
y(x)=\sumckxk (k=0 to ∞)
y'(x)=\sum(kckxk-1) (k=1 to ∞)
y''(x)=\sum(k(k-1))ckxk-2 (k=2 to ∞)
The Attempt at a Solution...
Homework Statement
Solve the following differential equation by power series and also by an elementary method. Verify that the series solution is the power series expansion of your other solution.
y'' = - 4y
Homework Equations
The Attempt at a Solution
y'' = - 4y \\
\frac{d^{2}y}{dx^{2}}...
Homework Statement
Length of rod = 1
Initial Conditions: u(x,0)=sin(πx)
Boundary conditions: u(0,t)=0 and u(1,t)=5.
Alright I am supposed to find the temperature at all times, but I am curious about the setup of the problem itself.
When x = 1, the boundary condition says...
Homework Statement
Find two power series solutions of the DE
(x+2)y'' + xy' - y = 0
about the ordinary point x = 0 . Include at least first four nonzero terms for each of the solutions.
2. The attempt at a solution
I distributed the y'' term and substituted
y = Ʃ0inf cnxn...
Homework Statement
Solve for y' = x^2y
The Attempt at a Solution
There's something that's been really bothering me about this question and similar ones.
We assume that the solution to the ODE will take the form
y = \sum_{n=0}{a_nx^n}
After finding y', plugging in the expressions...
Homework Statement
(For Physics 306: Theoretical methods of Physics... Text book: Mathematical Tools for Physics (really good!)...
Assumption: 300 level class = considerably junior level class)
-- Find a series solution about x=0 for y''+ysec(x) = 0, at least to a few terms.
(Ans...
Homework Statement
The question is y' - xy = 0
I have to solve it using series solutions
Homework Equations
The Attempt at a Solution
I use y = Ʃ from 0 to infinity a_n x^n and took the derivative. I plugged it into the equation
I got the recurrence relation to be a1 = 0...
I can't for the life of me figure out where C_{0} went. I scanned my work here..I've looked back and forth through my book and other texts, it always seems like all the coefficients are accounted for and/or they equal zero. As it stands, I only have C_{1} Thanks!
Full size...
Series solution for y"+x*y=0
Working on recurance realtion.
Get to (sum(n=2))n*(N-1)*a(n)*X^(n-2)+(sum(n=0))a(n)*x^(n)
Try several things but not sure if their correct.
Homework Statement
Consider the PDE
which has the solution
The Attempt at a Solution
So what I am having trouble is solving it using this method.
I am going to say that my $$u(x,t) = \sum_{n=1}^{\infty} u_n(t) \sin(nx)$$ and $$x \sin(t) =...
Hey!
I'm having problems with finding the general solution of this DE,
using series.
I have readed the Zill book, but I don't know how to start solving.
Any help is appreciated!
y'' - 4xy' -4y = e^x
Homework Statement
Find the indicial equation and find 2 independent series solutions for the DE:
xy''-xy'-y=0 about the regular singular point x=0
Homework Equations
y=Ʃ(0→∞) Cnxn+r
y'=Ʃ(0→∞) Cn(n+r)xn+r-1
y''=Ʃ(0→∞) Cn(n+r)(n+r-1)xn+r-2
The Attempt at a Solution
Finding the...
1. "Homework Statement
Find a recurrence formula for the power series solution around x = 0 for the differential equation given in the previous problem."
The previous problem says:
"Determine whether x = 0 is an ordinary point of the differential equation y'' + y = 0."
Homework...
Homework Statement
Determine φ''(x0), φ'''(x0), and φ(4)(x0) for the given point x0 if y=φ(x) is a solution of the given initial value problem.
y'' + (sinx)y' + (cosx)y = 0 y(0) = 0; y'(0) = 1
Homework Equations
y = φ(x) = Ʃan(x-x0)n
The Attempt at a Solution
I started off by...
Hi there. I have this differential equation: x^4y''+2x^3y'-y=0
And I have to find one solution of the form: \sum_0^{\infty}a_nx^{-n},x>0
So I have:
y(x)=\sum_0^{\infty}a_n x^{-n}
y'(x)=\sum_1^{\infty}(-n) a_n x^{-n-1}
y''(x)=\sum_2^{\infty}(-n)(-n-1) a_n x^{-n-2}
Then, replacing in the diff...