Tagged Questions
0
votes
1answer
24 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
34 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
33 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 ...
0
votes
0answers
43 views
Solutions of Chebyshev equation about $x = 1$
I need to find two solutions of the Chebyshev equation: $(1-x^2)y'' - xy' + a^2y = 0$, where a is some constant. Attached is a picture of what I have so far. I apologize if its kind of hard to read. ...
1
vote
0answers
76 views
Relating terms in differential equation with power series
Having problems with a task on a differential equation containing a power series.
Given
$$\frac{dx}{dt} = \lambda x + \sum_{n=2}^\infty b_n x^n$$
$$\frac{dy}{dt} = \lambda y$$
$$x(y) = y + ...
1
vote
2answers
62 views
Index of summation shift
I'm learning about power series in Differential Equations.
Right now I'm learning about shifting summations and something that is bothering me is the following:
Take the equation
$$F(x) = (x-3)y' + ...
1
vote
1answer
79 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
48 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
114 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
$$w_{n+2}=\frac{r_{n+2}-(n+1)p_0w_{n+1}-\sum_{k=0}^n w_k ...
0
votes
2answers
55 views
Using Eulers equation to find General Solutions
I have the problem $y''+4y = 0$ where $y(0) = 1$ , and $y'(0) = 0$
I have to find a particular solution, making a guess it is in the form $y(x) = e^{rx}$
I have the solution here, but I cannot quite ...
1
vote
1answer
70 views
Analytic Function without Power Series
If $$f(x) =\sum_{n=0}^{\infty}x^ n$$ Then Determine the function $f(x)$. Discuss the domain of $f(x)$. Discuss the domain of the derivative of $f(x)$. Thanks!
2
votes
2answers
61 views
An approximate solution to an ODE
I am interested in the ODE:
$x^\prime = x^2 + t^2$
$x(0)=0$
The power-series method is not (easily?) applicable here. Do you have any suggestions how to solve it?
2
votes
1answer
82 views
Verify that the equation $y''+y'-xy=0$ has a three-term, recursion formula and find its series solutions..
Verify that the equation $y''+y'-xy=0$ has a three-term, recursion
formula and find its series solutions $y_1$ and $y_2$ such that
$a)$ $y_1(0)=1$, $y_1'(0)=0$;
$b)$ $y_2(0)=0$, ...
2
votes
1answer
138 views
How to find the general solution of $(1+x^2)y''+2xy'-2y=0$. How to express by means of elementary functions?
Find the general solution of $$(1+x^2)y''+2xy'-2y=0$$
in terms of power series in $x$. Can you express this solution by means
of elementary functions?
I know that $y= ...
2
votes
2answers
176 views
Solving differential equation with power series
$$\begin{cases} w''=(z^2-1)w \\ w(0)=1 \\ w'(0)=0 \end{cases}$$
I tried the following: Let
$$w(z)=\sum_{j=0}^{\infty}w_j z^j$$
$$\implies w''(z)=\sum_{j=0}^{\infty}j(j-1)w_j z^{j-2}$$
$$\implies ...
2
votes
1answer
74 views
Show that the series representation of the Bessel function works
For the following series representation of the Bessel function:
$$w = J_n = \sum_{k=0}^{\infty} \frac{(-1)^k z^{n+2k}}{k!(n+k)!2^{n+2k}}.$$
I want to show that w is indeed the Bessel function, such ...
0
votes
2answers
67 views
How to find $a_0, a_1, a_2$ in a power series for an initial value problem?
Assume $y=\sum a_n x^n$. The ODE is $$y'' + (2 - 4x^2)y = 0$$
$y(0) = 1, y'(0) = 0$
$a_0 = 1, a_1 = 0, a_2 = -1$
2
votes
1answer
467 views
Frobenius Method to solve $x(1 - x)y'' - 3xy' - y = 0$
So, Im trying to self-learn method of frobenius, and I would like to ask if someone can explain to me how can we solve the following DE about $ x = 0$ using this method.
$$ x(1 - x)y'' - 3xy' - y = 0 ...
3
votes
0answers
121 views
Chebyshev Diff EQ
Find a power series solution about $x_0=0$ for the Chebyshev differential equation
$$(1-x^2)y''-xy'+n^2 y=0,$$
as a function of of the integer $n$. Show that the solutions form a terminating ...
2
votes
1answer
96 views
How can I express such function as known functions or power series?
$$\int_0^x \cfrac{1}{1+\int_0^t \cfrac{1}{2+\int_0^{t_1} \cfrac{1}{3+\int_0^{t_2} \cfrac{1}{\cdots} dt_3} dt_2} dt_1} dt =f(x)$$
$$\int_{0}^{x} \frac{1}{n+h_{n+1}(t)}{d} t=h_n(x)$$
...
0
votes
0answers
41 views
Solving for coefficient from Series
Where m = ω = h = c1 = x0 = constants
I'm trying to solve a second order differential equation with non-constant coefficients and I using series to solve for it. I've plugged in the series solution ...
2
votes
2answers
439 views
Help on differential equation $y''-2\sin y'+3y=\cos x$
$y''-2\sin y'+3y=\cos x$
I'm trying to solve it by power series, but I just can't find the way to get $\sin y'$. Is there any special way to find it?
1
vote
0answers
107 views
How to derive to inverse z transform of $\sqrt{\frac{1-a^2}{1-\frac{a}{z}}}$ from Laguerre differential equation?
How can I derive the inverse z-transform of:
$$\sqrt{\frac{1-a^2}{1-\frac{a}{z}}}$$
If Maple is not the way, how to derive manually?
With Maple code I encounter some problems
...
2
votes
0answers
79 views
Solve $x^2u''+xu'-(x^2+\frac{1}{4})u=0$ using power series
I stumbled upon this question in an old exam (I'm preparing for an exam of a course about ODEs). I didn't have much difficulty solving the Legendre and Hermite equations using power series, but this ...
1
vote
1answer
146 views
Series Solution Near Ordinary Points for Second Order Differential Equations
Given $(1+x^2)y''+2xy'-2y = 0$
The above equations obviously has analytic points everywhere except for $x=1$ and $-1$.
Find two linearly independent solutions $y_1$ and $y_2$ to the differential ...
1
vote
1answer
291 views
Squaring an arbitrary summation?
I'm trying to find a recurrence relation for the coefficients for the Maclaurin series for $\tan(x)$ by substituting $y=\sum_{k=0}^{\infty}C_{2k+1}x^{2k+1}$ into the differential equation $y'=1+y^2$. ...
6
votes
2answers
166 views
Power series $x f''(x) + f'(x) + xf(x) = 0$
Find a power series with radius of convergence $R = \infty$ such that $f(x) = \sum_{n=1}^{\infty} a_{n}x^{n}$ satisfies $x f''(x) + f'(x) + xf(x)= 0, \forall \mbox{ } x \in \mathbb R$.
How should ...
2
votes
2answers
210 views
How to solve the ordinary differential equation?
$\displaystyle\frac{d²y}{dx^2}+ \frac{4}{y}\left(\frac{dy}{dx}\right)^2+2=0$
with $y(0) = 1$ and $\displaystyle\frac{dy}{dx} = 0$ for $x = 0$.
1
vote
2answers
120 views
Solving $y'' - xy'+(3x-2)y=0$ using power series
I am trying to solve this equation using the series $$\sum_0^\infty a_nx^n$$
$$y'' - xy'+(3x-2)y=0$$
How to do that? I mean that I can replace the variables using the series but then I ...
2
votes
2answers
108 views
Solving a Second Order Linear Equation with Power Series
$$
y''+y'+xy=0
$$
I can't seem to get $y_1$ or $y_2$ to have any sort of pattern. I understand the technique to it but have no idea what the general solution is.
The equation I have ...
2
votes
1answer
829 views
A series solution of the differential equation: $\frac{d^2u(x)}{dx^2}+ u(x)^n = 0.$
Consider the differential equation
$\frac{d^2u(x)}{dx^2}+ u(x)^n = 0.$
Let the solution be
$u(x) = u_0(x) + p u_1(x) + p^2u_2(x) + \cdots +p^m u_m(x).$
Now we are interested in substituting the ...
3
votes
0answers
199 views
Series of nested double integrals
This is kind of a follow-up of my previous question. I'm investigating the following infinite series of nested two-dimensional integrals
$$\sigma(t,t^\prime) = 1 - \int_{t^\prime}^t\mathrm dt_1 ...
