0
votes
1answer
19 views

prove that $f(x,y) = x^2+y^2$ is continuous on rectangle R.

where $R = \{(x,y): |x|, |y| \leq \frac{1}{\sqrt 2} \}$ I am trying to use picard's theorem so I have to prove that f is continuous on R and that it's lipschitz continuous. How would I do this? I ...
0
votes
0answers
15 views

Fourier series and Legendre polynomials

I am currently dealing with a problem that is based on this question that I cannot answer and therefore I wanted to ask you for help on this simpler problem: Consider the Legendre differential ...
0
votes
0answers
16 views

Solution of differential system

Consider the following differential system in $\mathbb{R}^{n}$: $$u'=Au+(x^{2}+1)^{-1}u---(1)$$ where $A$ is an $n\times n$ matrix with real entries such that all eigenvalues of $A$ have ...
1
vote
3answers
44 views

Integration in question could not be resolved.

I do not know how to solve this integration
2
votes
0answers
25 views

How to define a Holder seminorm of a section

I'm reading "Variational Problems in Geometry",Seiki Nishikawa, in the figure below. Let $(M,g)$ be a compact $m$ dimensional Riemannian manifold with no boundary. $T>0, 0<\alpha<1, ...
2
votes
3answers
93 views

Find a particular solution of $\,\,y''+3y'+2y=\exp(\mathrm{e}^x)$

I already solved for the homogeneous one, but I'm still looking for the particular solution of the differential equation: $$y''+3y'+2y=\exp(\mathrm{e}^x)$$ The homogeneous solutions of this system ...
2
votes
1answer
59 views

Solving the ODE $\,\,x^4yy''+x^4(y')^2+3x^3yy'-1=0$

I'm currently trying to solve the differential equation $$x^4yy''+x^4(y')^2+3x^3yy'-1=0$$ I've tried the substitution $$v=\frac{y}{x}$$which didn't simplify the whole lot. Then I tried rewriting it ...
1
vote
1answer
28 views

Second order linear ODE $y^{\prime\prime}+\frac{2y^{\prime}}{x}-\frac{2y}{x^2}=0$

I have $y^{\prime\prime}+\frac{2y^{\prime}}{x}-\frac{2y}{x^2}=0$ How do I solve this? What have I tried? $1)$ Coupled system: $\begin{pmatrix}y_1^{\prime} \\ ...
1
vote
1answer
20 views

Change of variable in differential equation legitimate?

Just a general question ( I don't want to solve this ODE, I just want to understand why this is legitimate to do or not): Assuming we have the ODE $$y'(x) - \cos(x) y(x)=0$$ on $[0,2\pi]$ Am I ...
1
vote
1answer
67 views

how to solve the system of differential equations for this particle?

I'm trying to solve this problem A particle of mass m moves under the action of gravity on the inner surface of a paraboloid of revolution $x^2+y^2=az$ which assumed frictionless. Obtain the ...
1
vote
3answers
45 views

Find solution of differential equation $y'(t)=-2y(t)+1$

Could you help me explain how to find the solution of the differential equation $$ y'(t)=-2y(t)+1, $$ with $$y(0)=1.$$ I know that the solution is $$y(t)=\frac12 (1+e^{-2t}).$$ How about the IVP ...
0
votes
2answers
37 views

How can I find the fixed points of this differential equation?

The problem is to find the fixed points for the equation: $ \ddot x + x + \alpha x^²= 0 $ (and then sketch the global flow of the equation) (for $\alpha>0$) I know that for the autonomous ...
3
votes
5answers
51 views

Linear independence of the functions $1,\cos(x),\cos(2x)$

I want to show that the functions $1,\cos(x),\cos(2x)$ are linearly independent in $C[-\pi,\pi]$. I computed the Wronskian determinat of these functions but at the points $x=0,-\pi,\pi$ the obtained ...
1
vote
0answers
49 views

Ode with step function in the right-hand-side

I want to solve the following ODE: $$\dot{X}(t,x)=F(X(t,x))$$ where $F(x)=1$ if $x>0$ and $-1$ if $x<0$. How to treat this discontinuous right-hand-side?
0
votes
1answer
21 views

An ODE with boundary conditions at infinity

I have a problem where: $\ddddot{x} - 2 \ddot{x} + x = 0$ With boundary conditions $x(0) = 1, \dot{x}(0) = 2, x(\infty) = 0, \dot{x}(\infty) = 0$ So I get my characteristic equation: $s^4 -2s^2 + ...
-1
votes
3answers
71 views

Is $f(x)=x$ the solution of an integral equation? [closed]

Suppose that $f:[0, \infty)\longrightarrow \mathbb{R}$ is continuous and $f(x) \neq 0 $ for all $x>0$. If $$ \big(\,f(x)\big)^2=2 \int_0^x f(t)\,dt, $$ for all $x>0$, is it then true that ...
0
votes
1answer
30 views

$2^{\mathrm{nd}}$ order nonlinear ODE: $4y''\sqrt{y}=1, y(0)=1, y'(0)=1$

I am solving this second order nonlinear equation, that is in the title. My solution is: $$ \frac{4}{3}(y^{1/2}+c)^{3/2}-4c(y^{1/2}+c)^{1/2}+a=x $$ where $c$ and $a$ are constants that spawned ...
3
votes
1answer
41 views

Gronwall type inequality

Is there a Gronwall-type inequality for bounding $u(t)$ such that $$\vert \partial_t u(t)\vert\leq C \{ u(t)+u(t)^\alpha\}$$ with $\alpha>1$ ?
1
vote
1answer
55 views

Autonomous differential equation

Let $f: \Bbb R \to \Bbb R$ and $x_0 \in \Bbb R$, such that $f(x_0)> 0 $, and assume that $x(t)$ is the solution of $x'=f(x)$, such that $x(0)=x_0$. If $f(x) > 0$ then $x(t)$ is defined for all ...
2
votes
1answer
128 views

Find $f$, such that $\,f,f',\dots,f^{(n-1)}\,$ linearly independent and $\,f^{(n)}=f$

I am trying to find a function $f\in\mathcal{C}^\infty(\mathbb{R},\mathbb{C})$, satisfying the differential equation $$ f^{(n)}=f, $$ and at the same time $\,f,f',\dots,f^{(n-1)}\,$ are linearly ...
5
votes
2answers
135 views

Picard Iterates Converge Uniformly

I have a homework question that asks to show that the Picard iterates $$ \phi_{n+1}(t) = \int_0^t 1 + \phi_n^2(s) \, ds, \quad \quad \phi_0(t) = 0 $$ converge uniformly on any compact interval $[-r, ...
1
vote
1answer
22 views

Vector Analysis (Parametized curve)

The question is find a familiar parameterized curve that has the property $r(t) \times\dfrac{dr}{dt}=0$. The only curve that I can see that works is the line through the origin. I was just wondering ...
0
votes
0answers
32 views

Why can't the general solution of separable first order ODE cross the stationary solution?

For example, if we have the following Cauchy problem: $y'=y^2-4, y(0)=0$ In class, our professor told us that $y=-2,2$ are the two stationary solutions, but how could it be, since our initial point ...
1
vote
0answers
94 views

Qualitative properties of solutions to a ordinary differential equation.

I have this problem : $$\begin{cases} -(p(t)u'(t))'=f(t,u(t))\\u(0)=u(+\infty)=0\end{cases}$$ we have that $u$ is continues, $f:\mathbb{R}^+\times \mathbb{R}\rightarrow \mathbb{R}$ is continuous and ...
5
votes
2answers
94 views

Green's function for $y''+y=f(x)$

This example is taken from the Wikipedia's article. Namely, find the Green's function for $$y'' + y = f(x)$$ with boundary conditions: $$y(0) = y(\frac {\pi} {2}) = 0.$$ The defining equation for ...
1
vote
0answers
50 views

Numerical analysis- Runge Kutta

I have: $$y'(x)= \sin(y); y(0)=1$$ I need to calculate the function values with Runge-Kutta. The range is [0,1]. My problem is that I need to choose the h (=dx) such that the error will be in order ...
0
votes
0answers
33 views

Using the Lyapunov-Perron method to find the local stable/unstable manifolds

Hello Stack Exchange community. I am currently having an issue finding the local stable/unstable manifolds of this system. After going at it for a few hours I believe the person who wrote this ...
0
votes
0answers
36 views

Integration of a differential equation

I've got some problems with integrating a ODE, so maybe someone could add some words of advice. Given the following equation: $z''(x)-2\gamma z'(x) +p(x)z(x)=0$, $(1)$ and $\varphi(x)=z(x)e^{-\gamma ...
0
votes
0answers
46 views

Show that a function is solution to differential equation

I have a homogenous differential equation $a_0 y'' + a_1 y' + a_2 y = 0$ and a function $y(t) = t e^{\lambda_0 t}$. First I am assuming that $\lambda_0$ is a root in the characteristic polynomial. ...
2
votes
1answer
91 views

How to solve such a nonlinear ODE, the analytical solution of which is known!

I have the following ODE with initial/boundary value conditions: $$\left. \begin{aligned} \left(x^2-10 x-y^2\right)y\, y'(x)+(x-5) y^2 y'(x)^2-(x-5) y^2=0 ;\qquad (\text{ODE})\\ y(0)^2=25;\qquad ...
0
votes
1answer
65 views

Particular solution of a system of linear differential equations

Let $A(t) \in \mathbb R^{2\times 2}$ and $b(t) \in \mathbb R^2$ continuous functions in an open interval $I$. Consider the system $$(1) \space X'=A(t)X+b(t).$$ Let $X_1,X_2$ be linearly independent ...
0
votes
1answer
27 views

About a Property of maximal solutions of separable ODE's $y'=g(x)h(y)$ for locally Lipschitz $h : U\to\mathbb R$, $U$ open

Theorem: Let $\varphi : (a,b) \to \mathbb R$ be a maximal solution of the IVP $$ y'(x) = g(x) \cdot h(y(x)), \quad y(x_0) = y_0 \quad (1) $$ with continuous functions $g : I \to \mathbb R$ and $h : U ...
0
votes
1answer
34 views

Question on a derivation regarding the non-linear ODE $x'' = -U'(x)$, $U$ potential

Let $U$ be a potential function, and consider the IVP $$ (*) \quad x'' = -U'(x), \qquad x(t_0) = x_0, \quad x'(t_0) = v_0. $$ We suppose the following: (V) Let $x_0, v_0$ be initial values and let ...
0
votes
0answers
18 views

Green's function to operator

I would like to understand how one can show that the Green's function in this table is a Green's function to the D'Alembert operator? I refer to the wikipedia page about Green's function
1
vote
0answers
33 views

Function satisfies differential equation.

Given the D'Alembert operator D'Alembertian $\Box$, I want to show that $$ G(x,t,x_0,t_0):= \frac{\delta \left(t_0 + \frac{||x-x_0||}{c} -t \right)}{||x-x_0||} $$ satisfies $$ \Box G(x,t,x_0,t_0) = ...
0
votes
0answers
22 views

Constant solutions of separable ODE

Consider the IWP $$ y'(x) = g(x) \cdot h(y(x)), \quad y(x_0) = y_0 $$ for continuous functions $g : I \to \mathbb R$ and $h : U \to \mathbb R$ on open intervals $I, U$ with $(x_0, y_0) \in I\times ...
0
votes
1answer
51 views

Linearizing systems about critical points.

$$\def\q{\begin{pmatrix}}\def\p{\end{pmatrix}}\def\l{\lambda}\def\f{\frac{\sqrt{11}}{2}}$$ Find all the critical points of the following systems and derive the linearised system about each ...
1
vote
1answer
91 views

Transforming ODEs into exact equations.

I want some examples of ODEs that can only be solved by transforming them into exact equations. They shouldn't be solvable with; Direct integration, separation of variables, manipulating a reverse ...
0
votes
5answers
111 views

Assumptions in Word Problems (Calculus)

I just had a small question about assumptions in mathematical word problems. Suppose you are given a calculus problem (related-rates), "A spherical balloon is inflated with gas at the rate of 800 ...
3
votes
1answer
34 views

Solution of differential equations with discontinuity

Suppose that we have scalar differential equation \begin{equation} \dot{x}(t)=u(t) \end{equation} Here $u(t)$ is a piecewise constant function with discontinuity. If the points of discontinuity is ...
4
votes
3answers
118 views

Solution of $\frac{d^2y}{dx^2} - \frac{H(x) y}{b} = H(-x)$

Does the equation $$\frac{d^2y}{dx^2} - \frac{H(x)}{b} y = c H(x)$$ have a solution where $H(x)$ is the Heaviside step function and $b$ and $c$ are constant? Update: What about the second step ...
2
votes
1answer
73 views

ODE $d^2y/dx^2 + y/a^2 = u(x)$

Does the following ODE: $$d^2y/dx^2 + y/a^2 = u(x)$$ have a solution? where $u(x)$ is the step function and a is constant.
0
votes
0answers
24 views

Integration of nonlinear and linear ODEs

\begin{equation} \frac{dc_1}{d\tau}= \alpha I(1-c_{0}) + c_{1} (-K_{F} - K_{D}-K_{N} s_{0}-K_{P}(1-q_{0}))+ c_{0}(-K_{N} s_{1}+K_{P}q_{1}), \nonumber \end{equation} \begin{equation} ...
0
votes
0answers
28 views

ODE Initial value problem formualtion

If I have the following ODE initial value problem, $$\begin{align} y'(t) &= f(t), \quad t>0, \\ y(0) &= y_0. \end{align}$$ Then I was taught that a solution to the problem is given by ...
1
vote
1answer
70 views

Existence of solution of ordinary differential equation

I am reading a proof of the existence of solutions for ordinary differential equations and I have some basic doubt. I'll copy the statement, the part of the proof I don't understand and my question: ...
4
votes
2answers
144 views

Solving a 2nd order nonlinear ODE

Could you help me solve or give me some advice about following differential equation $$ 2(y')^2 + 3xy'y'' + 3yy'' = 0 $$
0
votes
2answers
42 views

Show Uniqueness of Solution for Boundary Value Problem

Let $G \subseteq R^n$ be a simple, connected and bounded region with smooth boundary and let $f : \overline G \to \mathbb R$, $g : \partial G \to \mathbb R$ be continuous. Show that the following ...
0
votes
1answer
30 views

Show that Fourier series arising in solution of differential eqn. converges uniformly

Let $f \in L_2(0,\pi)$ have the Fourier expansion $f(x) = \sum_{n=2}^{\infty} f_n\sin(nx)$. Compute (formally) the boundardy value problem $$ u''(x) + u(x) = f(x) \qquad \mbox{ for } 0 < x < ...
0
votes
1answer
41 views

Prove two solutions of differential equation are the same

In a recent work I had to solve the following differential equation: $$ r x''(r)+r x'(r)^2+x'(r)-\frac{4}{r}=0~~. $$ To do so I used two methods and I got, using each, two solutions with different ...
1
vote
0answers
57 views

existence and uniqueness of volterra integral equation of the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...