Tagged Questions
0
votes
1answer
38 views
Question about eigenvalues
I have this :
i dont understand why they write $\lambda=m^2 , m\in \mathbb{N}\cup\lbrace0\rbrace$ ,
it's right that $\lambda=m^2$ is the eigenvalues of $(P_0)$ ,but $0$ is not an eigenvalue !.
...
2
votes
2answers
36 views
Finding a strong enough solution to a specific PDE problem.
Let $U\subset \mathbb{R}^n$ with smooth boundary $\partial U$. And consider the expression
$$\Delta u = f.$$
$$\text{+"convenient boundary conditions"}$$
In my specific case $f\in H^2_0$. Under ...
1
vote
0answers
48 views
Approximating the modified Bessel’s function with a sum of exponentials
I am looking for an approximation for modified Bessel’s function $I_\alpha(f(t))$ (specially $I_0(f(t))$ or at least $I_0(t)$) with a sum of exponential functions. I mean I want to approximate the ...
4
votes
2answers
53 views
Does this ODE have an exact or well-established approximate analytical solution?
The equation looks like this:
$$\frac{\mathrm{d}y}{\mathrm{d}t} = A + B\sin\omega t - C y^n,$$
where $A$, $B$, $C$ are positive constants, and $n\ge1$ is an integer. Actually I am mainly concerned ...
2
votes
1answer
72 views
Possible ways to do stability analysis of non-linear, three-dimensional Differential Equations
For example Lorenz system,
$$
\frac{d}{dt}\begin{pmatrix}
x\\
y\\
z
\end{pmatrix}=\begin{pmatrix}
-\sigma & \sigma & 0\\
\rho & -1 & -x\\
y & 0 & -\beta
...
1
vote
0answers
38 views
Pull Back (change of variables)
Let be $h:\mathbb{R^2}\rightarrow\mathbb{R^2}$ a change of variables (diffeomorphism). Let be $X$ a vector fields in $\mathbb{R^2}$ and $f:\mathbb{R^2}\rightarrow\mathbb{R}$ a continuous application. ...
2
votes
1answer
37 views
Unique solution differential equation proof
Prove that there is a $\delta>0$ such that there is a unique solution of the differential equation $y'(t)=\sin(y(t))$ with $y(0)=1$ on the interval $[-\delta, \delta]$. How large can you choose ...
1
vote
1answer
84 views
Partials and maximization
If we have that the contours of a response surface are elliptical and the response is given by the following function:
$$\large \exp\left(-\left(w^2 + \frac{1}{4}l^2 -\frac{1}{4} \cdot w \cdot ...
0
votes
1answer
30 views
Behaviour of solution of autonomous ODE
I need to determine whether the solution to the ODE
\begin{equation}
y^\prime = -x\exp(\frac{y^{2\alpha}}{2}), \qquad \alpha \in (0,1)
\end{equation} is such that $y = o(\frac{1}{\sqrt{x}})$ as $x \to ...
0
votes
0answers
22 views
DFT in complex form and Trigonometric Polynomial Interpolation: why different dimensions of basis vectors set?
I'm trying to figure out, how coefficients of Discrete Fourier Transform in complex form are converted into coefficients of Trigonometric Polynomials Interpolation:
Say, I have a function vector with ...
1
vote
0answers
33 views
Importance of Riemann-Liouville fractional derivative from historical point of view
Why Riemann-Liouville fractional derivative is important from historical point of view than that of Caputo fractional derivative? As we know Riemann-Liouville fractional derivative is more theoretical ...
0
votes
1answer
33 views
Show $y''+\lambda^2y=f(x)$ has root : $y(x)$ satisfying condition : $y(0)=y'(0)=0.$
Show that the equation : $y''+\lambda^2y=f(x)$ has root : $y(x)$ satisfying condition : $y(0)=y'(0)=0$ and $$y(x)=\frac{1}{\lambda}\int_{0}^{x_0}\sin\lambda(x-\tau)f(\tau)d\tau$$
0
votes
0answers
48 views
About calculating a Green's function
For a positive integer $d>=2$ and a real number $h_0$ I have the differential equation for a function $f$ of $x$,
$x^2 f'' + h_0^2 x f' = \frac{d(d-2)}{2} f$
The eigenfunctions are then $f \sim ...
2
votes
3answers
47 views
Solving $(f'(x))^2 = f(x)f''(x)$ with boundary conditions.
Let $f$ be a continuous real-valued function such that $$(f'(x))^2 = f(x)f''(x).$$ Suppose $f(0) = 1$ and $f^{(4)} (0) = 9$. Find all possible values of $f'(0)$.
I have this question in my book ...
4
votes
3answers
108 views
Proving Newton's Binomial Theorem
So, I've done most of the problem to this point, but just cannot figure out the last piece. I may just be missing the math skills needed to complete the proof (differential equations).
Problem (from ...
1
vote
1answer
94 views
$y''(t)=-g+y'(t)^2/y(t)$ unique solution
I am looking for a theorem of proof that tells me, that
$$
y(t)=r \left( 1-\cos \left( \sqrt{\frac{g}{r}}t \right) \right)
$$
is a unique solution to the differential equation
...
0
votes
0answers
17 views
Numerical methods for ODE
(excuse my english in advance)
I have an ODE $$\dot{\alpha}(t,x)=V(t,\alpha (t,x),\lambda)$$
$$\alpha(0,x)=x$$ where the field $V$ is $C^{\infty}$ with respect to $t$,$x$ and a vector of parameters ...
2
votes
1answer
85 views
Looking for help with a proof that n-th derivative of $e^\frac{-1}{x^2} = 0$ for $x=0$.
Given the function
$$
f(x) = \left\{\begin{array}{cc}
e^{- \frac{1}{x^2}} & x \neq 0
\\
0 & x = 0
\end{array}\right.
$$
show that $\forall_{n\in \Bbb N} f^{(n)}(0) = 0$.
So I have to show ...
1
vote
1answer
47 views
Boundary-value problem in differential equations
Consider the problem:
$$u^{(4)} + \lambda u = 0, \ \ \ 0<x<\pi; \ \ \ u(0) = u(\pi) = u''(0) = u''(\pi) =0$$
Find the eigenvalues.
How should one proceed about this problem? I am complete ...
0
votes
0answers
30 views
Linear operators and Green's function
Consider the operator $Lu= -u'$ and the boundary condition $\alpha u(a) + \beta u(b) = 0$. Under what conditions on $\alpha, \beta$ is it true that the boundary-value problem given below has only the ...
0
votes
1answer
84 views
Solution of second order differential equation
Consider the differential equation:
$$u''(x) = \lambda \ u(x), \ \ \ 0 <x < \pi$$
I want to find solutions $u_1, u_2$ such that the following data are satisfied:
$$u_1(0) = 0, \ \ u_1'(0) = ...
0
votes
0answers
30 views
Mixed-endpoint boundary value problem
Consider the following mixed-endpoint boundary-value problem:
$$- \displaystyle \frac{d}{dt} \left (p(t) \frac{du}{dt} \right ) + q(t)u = \lambda u, \ \ \ \ u(0) = u(T), \ u'(0) = u'(T)$$
here the ...
-1
votes
1answer
293 views
Topological degree
I need help for this exercice
1)Let $\Omega$ be an open and bounded set from $\mathbb{R}^n$ and $f\in C(\overline{\Omega})$
,we suppose that there exists
$x_0 \in \Omega$ such that :if for $x\in ...
1
vote
2answers
105 views
Stability analysis, or, Can we prove this limit to be zero?
Let's think about this ODE
$$
\dot{y}(t) = \gamma \left(g(t) - y(t)\right),\quad \gamma > 0,
$$
where $g(t)$ is a Lipschitz continuous function. It can be seen that the value of $y(\cdot)$ goes ...
3
votes
2answers
264 views
$y''\pm e^ty=0 \implies \mid \cup x_i \mid =? s.t. y(x_i)=0 $
I have this question, and i don't know how to solve it:
Show that the solutions of $y''+e^ty=0$ admit an infinite number of zeros.
Also, how to prove that the solutions of $y''-e^ty=0$ admit not ...
1
vote
1answer
72 views
Picone formula (Sturm oscillation)
We consider the equations:
$(p_1y')'+q_1y=0 ...(E_1)$ ,$(p_2y')'+q_2y=0 ...(E_2)$
$p_1,p_2 \in C^1([0,1],(0,\infty)) ; q_1,q_2 \in C([0,1],\mathbb{R})$
$y_1$and $y_2$ are respectivly solutions of ...
1
vote
1answer
26 views
Necessary conditions for transforming a system of O.D.Es to a single O.D.E.
Considering the system:
$$u' = a(x)u + b(x)v$$
$$v' = c(x)u + d(x)v$$
I transform it to the second order O.D.E.(please check me on this because I might have mistakes):
$$\displaystyle u'' - \left ...
3
votes
1answer
36 views
Finite family of analytic functions linearly dependent if and only if Wronskian is 0
I know that given two analytic functions on some domain $D$ of the complex plane, then their Wronskian determinant being $0$ is equivalent to them being linearly dependent. I would like to generalise ...
1
vote
0answers
22 views
Boundaries- regularity and local parametrization
Suppose we have a bounded domain $\Omega \subset \mathbb{R}^3$ with $C^2$ boundary.Let $x_0 \in \partial \Omega$. We choose a $X_1,x_2,x_3$- coordinate system such that the $x_1,x_2$-plane is ...
0
votes
1answer
25 views
What is $| g(t,x) |$ for multidimensional $g$?
I'm reading a book on ODE, and find $|\cdot|$ is confusing. It says:
Consider a function $g:\Omega \rightarrow \mathbb{R}^n$. For every compact $K\subset \Omega$, there exist constants $C$ and $L$ ...
0
votes
1answer
48 views
A change of variables in the euler equation
If someone could help me with the proposed change of variables, it would be greatly appreciated. Consider Euler's equation:
$$z^2w'' + \alpha zw' + \beta w = 0$$
where $w$ is a function of $z$ and ...
1
vote
1answer
82 views
Legendre's equation polynomial solution
This is a problem on analytic solutions of ordinarry differential equations. Any help will be greatly appreciated. Please, try to be as specific as possible as I don't handle this material very well ...
1
vote
3answers
79 views
Linear instability implies nonlinear instability
I am trying to understand the following proof that linear instability implies nonlinear instability. Suppose we have the ODE, $ \frac{du}{dt}=A(u)$ for which $0$ is a solution. Suppose $L $ is the ...
1
vote
2answers
103 views
Solving a second order inhomogeneous differential equation with constant coeffcients
I a seeking to solve the equation: $$u'' + 2vu' + u = cos(\sigma t), \ u(0) = 1, \ u'(0) =0$$ where $0 < v < 1$. Then I have to show that the solutio is purely oscillatory (which I don't know ...
2
votes
1answer
77 views
Help with Initial value problem : $y'= x^2+ xy^2, y(0) = 0$; Picard–Lindelöf Approximation.
i need solve this: $$y'=x^2+xy^2 , y(0)= y(t_0)= 0$$
a) Compute, starting from the constant function $u_0=0$ the successive approximations $u_1,u_2,u_3$ (in the sense of the theorem of ...
1
vote
1answer
85 views
Lipschitz condition on a second order nonlinear ODE?
Preliminaries:
Let the matrix norm be $$\sqrt{\sum_{j=1}^n\sum_{i=1}^n a_{ij}^2}=||\mathbf A||.$$
I am trying to prove uniqueness and existence of a second order nonlinear ODE (Ordinary Differential ...
5
votes
2answers
100 views
Estimating rate of blow up of an ODE
Suppose I have a differential equation $x'=f(x)$ and $f(x)>0$ grows super-linearly. I.e., $\lim_{|x| \rightarrow \infty} |f(x)|/|x| \rightarrow \infty$.
Several related questions: (1) Can I ...
0
votes
0answers
48 views
How to derive a differential equation from a relation with differentials
Suppose I have a function of time $t$ and position $(x,y)$ such that
\begin{equation} p_t \,dt = p \,dy - p_x (1-x) \,dx + p_y \,dy\end{equation}
where the subscript denotes a differentiation. In this ...
0
votes
1answer
308 views
General Solution to Quasilinear PDE using Method of Characteristics
This is a homework that I'm having a bit of trouble with. I posted it previously but there was a typo in my original post. Since I received an answer for the incorrect problem it was suggested that ...
0
votes
2answers
128 views
Vector field with bounded integral curves
I am thinking about smooth vector fields on some (open set of an) euclidean space $\mathbb{R}^n$.
I know that the integral curves of a general vector field $X$ are not defined for every time $t\in ...
0
votes
1answer
28 views
Information needed about Local Extremas in differential calculus
We know a function $f \in C^2(R)$ has a Local Maximum in the origin $(0,0)$. What can you say about the differential: $d_{(0,0)}^2f(1,-1)<0$?
I've recently got this on a test and I'm not sure if ...
3
votes
1answer
112 views
Second order linear ODE with variable coefficients
Consider the second-order linear differential equation $u'' + p(x)u' + q(x)u = 0$ where $p$ and $q$ are continuous on the entire $\mathbb{R}$. Suppose that $q(x) < 0 $ everywhere. Show that if $u$ ...
0
votes
1answer
78 views
Solution to system of linear ODEs backwards in time
Suppose we have a system of linear ODEs represented in state-space form as follows:
$$ \dot{x}(t) = Ax(t) + Bu(t), \; x(t_0)=x_0. $$
The well-known solution, valid for $t\in[t_0,\infty)$, is
$$ x(t) = ...
1
vote
2answers
88 views
An easy partial differential equation
I have just entered the study of ODEs. However, the professor, without having talked at all about it in class, asked us to solve the following partial differential equation:
$\displaystyle ...
16
votes
5answers
679 views
If $f''(x)+f(x)>0$ and $f(x)>0$ $\forall x\in(a,b)$; $f(a)=f(b)=0$; prove that $b-a>\pi$.
Please help me to solve this question:
Suppose $f:[a,b] \to \Bbb R$ satisfies:
$f''(x)+f(x)>0$ and $f(x)>0$ for all $x\in(a ,b)$;
$f(a)=f(b)=0$.
Prove that $b-a>\pi$.
...
2
votes
1answer
158 views
Similar proof of Peano's Existence Theorem
As many of you will know, Peano's theorem states that if $f(x,y)$ is continuous and bounded in the strip $T: |x-x_0| \le a, |y|\le\infty $. Then the intitial value problem $y'=f(x,y), y(x_0)=y_0$, ...
4
votes
1answer
178 views
Method of isoclines
I have this exercise and I do not know how to solve it.
By using the method of isoclines represent the integrals of equation corbes nonautonomous $x'=x^2-t$.
There are some indications: Let $P = ...
0
votes
1answer
42 views
The property of ODE $y''+(1+\cos{x})y'+xy=\sin{x}$
$y''+(1+\cos{x})y'+xy=\sin{x}$ with $y(0)=2$.
Is $y(0)$ the local maximum or minimum? What is the convexity near $x=0$ (in a region $(\epsilon,0)$ with $\epsilon$ small enough, is y(x) convex or ...
5
votes
3answers
157 views
How do I show there are no elementary function solutions for the differential equation $f''(x)=f(\sqrt{x}), x>0$?
How do I show there are no elementary function solutions for the differential equation $f''(x)=f(\sqrt{x}), x>0$ in the $C^2(0,\infty)$ space solutions?
0
votes
1answer
187 views
Initial value problem: Solution blows up conditions
Consider the differential equation in $\mathbb R$:
$$x' = x^2-\lambda x^3; \,\,\,\,\,\, x(0) = x_0; \,\,\,\,\, t\geq 0$$
where $\lambda $ is a parameter. For which initial conditions is the solution ...
