Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.

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2
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3answers
55 views

Show uniqueness of solution to $\dot{x}(t) = \frac{x}{2+2t}$

In a homework problem, the diff EQ came up $$\dot{x}(t) = \frac{x(t)}{2+2t}$$ and $x(1) = 1$. I note that $x(t) = \frac1{\sqrt{2}}\sqrt{1+t}$ is a solution. How might I show uniqueness of the ...
1
vote
1answer
74 views

differential equations and physical intuition

Often when you study differential equations, you find phenomena in nature modeled by those equations. Sometimes an insight into a physical problem can help you to solve a differential equation. My ...
1
vote
0answers
19 views

Trace of a tensor in a differential equation

If $Z$ is a rank-2 tensor, does the following differential equation mean anything to anyone: $\nabla^2Z+\frac{1}{c^2}\frac{\partial^2}{\partial t^2}tr(Z)=0$ The presence of this trace really blurs ...
9
votes
1answer
75 views

Set of points dense in subset of four-dimensional space

We may assume the following theorem: Theorem: A real number $\lambda$ is irrational iff the set $\{m+\lambda n\mid m,n\in\mathbb{Z}\}$ is a dense subset of $\mathbb{R}$. Consider the following ...
5
votes
0answers
99 views

Solving numerically the equation of motion of D7 brane perturbation

I want to solve this equation $$ \partial_{\rho}^{2}\phi+\frac{3}{\rho}\partial_{\rho}\phi+\left(\frac{M^{2}}{(1+\rho^{2})^{2}}-\frac{l(l+2)}{\rho^{2}}\right)\phi=0 $$ numerically. I know that ...
0
votes
0answers
76 views

Simplify sum and solve for r

I have the follow so far, and now I'm stuck at solving for r. I know r=1, r=3. The context behind this problem is, given a second-order d.e. use forbenius' method to solve for the indicial roots and ...
0
votes
1answer
81 views

Solving Second order ODE with variable coefficients?

$m\ddot{x} + c(x)\dot{x} + k(x)x = 0$ where $\dot{x} = \dfrac{dx}{dt}, \ddot{x} = \dfrac{d^2x}{dt^2}$ and $k(x), c(x)$ are functions of $x$. I saw some methods to solve variable coefficient ODEs but ...
1
vote
1answer
41 views

Problem with solving non-linear differential equation.

Need some hints where to start with this non-linear differential equations. $$\ddot{r} = \dot{r} (\dot\varphi)^2 - \frac{2rk}{m}$$ $$\ddot\varphi=-\frac{\dot{r}\dot\varphi}{r}$$ Thanks in advance ! ...
0
votes
1answer
36 views

Ordinary Differential Equations with initial conditions

Let $y_1(x)$ and $y_2(x)$ be the solution of $y' = y+17$ with $y_1(0)=0$ and $y_2(0)=1$. Then (a) $y_1$ and $y_2$ will never intersect (b) $y_1$ and $y_2$ will intersect at $x = 17$ (c) $y_1$ and ...
3
votes
2answers
311 views

Examples of applications of Linear differential equations to physics.

I wonder which other real life applications do exist for linear differential equations, besides harmonic oscillators and pendulums. I'm looking for examples to include in a document that talks about ...
2
votes
0answers
61 views

On a specific non-linear partial differential equation

Given an $n$-dimensional variable $\mathbf{x}\in\mathbb{R}^n$ and the functions $h_i: \mathbb{R}^n \rightarrow \mathbb{R}$, $i=1,\dots,l$, we would like to find a solution of the following equation: ...
0
votes
1answer
165 views

Binomial theorem at power series expression

Given the binomial theorem of $(1+x)^n$ power series expression as $\sum_{k=0}^n {n\choose k} x^k = (1 + x)^n$. Use the binomial theorem and find a power series expression of the from ...
2
votes
2answers
50 views

Does “$\exists \delta >0$ S.T $||x(0)-x_e||<\delta\Rightarrow \displaystyle \lim_{t\rightarrow \infty}||x(t)-x_e||=0$” imply stability?

Recall the definition of stable and Asymptotically stable: A fixed point $x_e$ of a vector field is called (Lyapunov) stable if $\forall \varepsilon>0,\exists \delta(\varepsilon)$ such that ...
3
votes
1answer
106 views

Prove $0$ is an exponentially stable equilibrium of the system $x'=f(x)+g(x)$ if $f(0)=g(0)=0$

Besides the conditions in the title, we have: $0$ is an exponential equilibrium of the system $y'=f(y)$ $|g(x)|\leq \mu|x|,\forall x \in \mathbb{R}^n$ $\mu$ is sufficiently small! What I have ...
1
vote
0answers
29 views

Change of variables FPE

Given the partial differential equation: $$\partial_tP(z,t)=-\partial_z[(-z^2+A)P(z,t)]+D\partial_{zz}P(z,t)$$ where $A$ and $D$ are constant parameters. how to remove $z^2$ term by substitution?
0
votes
1answer
39 views

Solution of linear differential equation help

$$df/dx=(f(x)+x)x, f(0)=0$$ Show that there is a unique solution to the dierential equation and find the solution. This may be linear differential equation but $f(x)$ only including $x$ so ...
1
vote
1answer
109 views

Change of Variables in differential equation

Given the equation $zZ''(z) + Z'(z) + \alpha^2Z(z) = 0$ use the change of variables $x = \sqrt{\frac{z}{a}}$ where $a$ is a constant to map the problem to the differential equation $Z''(x) + ...
1
vote
1answer
46 views

$(D^2 -1)y=0 \iff (D-\tanh x)(D+ \tanh x)y=0$ where $D \equiv \frac{d}{dx}$?!

In this lecture, the professor explains that factoring differential operators if different from factoring numbers, and provides this example. I have no idea where it comes from and indeed why it is ...
0
votes
1answer
73 views

2nd order differential equation

I'm working on the following 2nd order ODE: $$ x^2 y''+2(2x-1/b^2) y'+2(1-(a/b)^2)y=0, $$ where $b\neq 0$. It's very similar to the equation for the generalized Bessel polynomials (see ...
3
votes
2answers
49 views

Show the system of differential equations has the solution of the form $ \phi(x)=e^{kx}\alpha$

Consider the following linear system $$y_1 '=ay_1+by_2$$$$y_2 '=cy_1+dy_2$$ where $a,b,c,d$ are constants. Then show that this system always has a solution $\phi(x)=e^{kx}\alpha,$ where ...
0
votes
2answers
39 views

Use change of variables to find solutions

Use change of variables to find solutions to $$ x^3 \frac{dy}{dx}=yx^2-y^3 $$ Which should I substitute?
1
vote
1answer
103 views

Solve the following Defferntial Equation $xy^2y'=\frac{x^3}{\ln\left(\frac{y}{x}\right)}+y^3$

I want to solve the following: $$xy^2y'=\frac{x^3}{\ln\left(\frac{y}{x}\right)}+y^3$$ I dont know if I can separate variables here because of the $\ln\left(\frac{y}{x}\right)$. Any suggestions? ...
0
votes
0answers
95 views

analytical solution for system of nonlinear ODEs in two dimensions

This ODE comes out of a system of self propelled particles (http://www.foelsche.com/swarm). Currently this system is linear -- I want to add a nonlinear friction/drag. Every particle has the following ...
1
vote
1answer
90 views

Counter-example to Cauchy-Peano-Arzela theorem

I was looking for a counter-example to Cauchy-Peano-Arzela theorem. I found this paper (in french) from Dieudonné. [acta.fyx.hu] Take $E = c_0$ to be the space of real sequences converging to $0$, ...
0
votes
2answers
78 views

Solving an initial value problem using Runge-Kuttas

I am trying to solve the equation Using Runge-Kutta methods, but the lack of y- values on the right hand side is confusing me. Any help would be much appreciated. My initial conditions are: dy/dx = ...
1
vote
1answer
41 views

Spherical Mean equivalent forms

I am having trouble understanding the following identity for 2 equivalent forms of the spherical mean: $$\dfrac{1}{d\omega_dr^{d-1}}\int_{\partial B(x, r)} v(y)do(y)=\dfrac{1}{d\omega_d} ...
0
votes
2answers
115 views

Elementary Function of Power series (e.x: exponential, logarithmic, Sine or Cosine )

a)Find the Unique Power series solution of, $y''+xy'+y=0$ given $y(0)=1$ and $y'(0)=0$ I have done the work and find the unique solution by using initial condition where the constant $a_0=1$ and ...
4
votes
2answers
2k views

Solve the ODE $yy'' + (y')^2 = 0$

I am asked in a book to solve the following ODE: $$y\dfrac{d^2y}{dt^2} + \left(\dfrac{dy}{dt}\right)^2 = 0$$ One solution for the ODE above is $y = 0$. I will use the following substitution: $$v = ...
0
votes
0answers
99 views

Adjoint of Second order differential operator.

So I have the following problem: Find the adjoint of the following operator: $u''+\pi^2u$, $u(0)=\alpha$, $u'(\frac{1}{2})=\beta$. To this purpose I get the following: ...
2
votes
1answer
64 views

Prove the following function is decreasing

Given: $E[u(t)] = \int_{\Omega} \left(\frac{1}{2}\left|\nabla u\right|^{2} - \frac{1}{4}u^{4}\right)dx$, and $u_{t} - \nabla u = u^{3}$. Show: $\frac{\partial}{\partial t}E\left[u(t)\right] \le 0$. ...
1
vote
1answer
75 views

Question from a conservation law example in Evans' PDE book

I'm trying to fill in some details in an example given in Evans' PDE book, chapter 3.4, example 1 on page 139. Starting with an initial-value problem for Burgers' equation: \begin{equation} ...
0
votes
1answer
34 views

When implies a linear relation in the function parameters also a linear relation in the derivatives?

Consider a function $f(x,y)$ of two variables $x$ and $y$. The variables $x$ and $y$ on which $f$ depends occur only in the form $a(x-by)$ with the constants $a$ and $b$. (For additional ...
0
votes
1answer
40 views

Mechanical oscillator and unbounded swings question

Will the mechanical oscillator with equation $$my′′+cy′+ky=f(t)$$ have unbounded swings using $$f(t)=\exp(−pt)\cosωt\quad ?$$ ($−p+iω$ is a root of $mr^2+cr+k=0$). I think the answer is yes, but ...
1
vote
1answer
46 views

Uniqueness/Existence ODE question

here's my question Consider the differential equation: $t\frac{dg}{dt} = 2g.$ I got that the general solution is $g = ct^2$. However, I don't understand how to answer these questions: What is the ...
0
votes
2answers
77 views

Construct the Green s function for the equation

Construct the Green s function for the equation y^''+ 2y^'+2y=0 Which boundary conditions y(0)=0 , y(π/2)=0 Is this Green s function symmetric? What is the Green s function, if the ...
1
vote
1answer
1k views

The number of solutions to an $n^{th}$ order differential equation.

For an $n$th order differential equation, why are there always $n$ solutions? Why exactly $n$, not $n - 1, n+1$ or infinite many? Addendum by LePressentiment : This is motivated by P176 on ...
2
votes
2answers
64 views

A way to show that if $y''(x) +y(x)=0$ then $y(x)=\cos (x)$

How can I show that if $y''(x)+y(x)=0$ then $y(x)=\cos(x)$. I found this out by intuition but is there a algebraic way to show this?
0
votes
1answer
45 views

linear homogeneous periodic equation

I'm having trouble with the following problem: Consider a linear homogeneous equation in the plane: $x'(t)=A(t)x(t)$ (1) Assume the matrix $A(t)$ has period $T$, in other words $A(t+T)=A(t)$. Show ...
6
votes
2answers
159 views

Initial Value Problem with Repeated Eigenvalues

Given the matrix $$ A=\left(\begin{array}{ccc} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 0 & 1\end{array}\right) $$ For $X'= AX.\quad$ $X\left(0\right)=\left(\begin{array}{r}1 \\ 0 \\ ...
1
vote
1answer
89 views

Laplace boundary value problem

I came across the following Laplace bvp: $u_{xx}+u_{yy}=0\ $ for $\ 0<x<1,\ 0<y<2$ $u(x,0)=u(x,2)=0$ $u(0,y)=0$ $u(1,y)=y(2-y)$ I didn't have any problems solving it. It was quite ...
-1
votes
1answer
122 views

ODE question about defining certain values

A parachutist whose mass is 100 kg drops from a helicopter hovering 3000 m above the ground and falls under the influence of gravity. Assume that the force due to air resistance is proportional to ...
1
vote
2answers
189 views

Change of Coordinate in Differential Equation

I'm sorry, it's probably a very simple question but I'm confused between change of variable and change of coordinate in a differential equation. To take a very simple example, let's start with this ...
1
vote
0answers
64 views

Lipschitz dependence on ODE parameters in discrete setting

Consider the ordinary differential equation $$ x'(t) = e^{-x(t)} - p$$ with (time-independent) non-negative parameter $p$. Its solution after time $t$ is given by $$ \Phi^t_px_0 = \log\left( e^{x_0 - ...
-1
votes
0answers
41 views

Numerical Solvers to deal simultaneously with very different types of Oscillatory Behaviour

I am trying to solve these two related problems numerically: \begin{align} &f^{(\mbox{v})}(y) -(f^5 (y))'-\frac{1}{6}yf(y)=0\\ f'(0)=f'''(0)=0, &\quad f(y) \sim Cy^{(-1/7)}\exp(\gamma ...
0
votes
2answers
220 views

Solutions to ODE: $y'=y^{1/3}$

I'm trying to find all tuples $(a,b,x_0,y_0)\in\mathbb{R}^4$ such that $a<x_0<b$ and there is a solution for the differential equation $y'=y^{1/3}$ on $(a,b)$ satisfying $f(x_0)=y_0$, and all ...
1
vote
1answer
41 views

Why isn't there a way to infer a formula from market data?

Looking the graphic of a stock, one could intuitively guess a mathematician could infer a formula that fit the graphic and, from that, get future values. Is that possible, to some extent, and what ...
3
votes
1answer
344 views

Lyapunov Stability of Non-autonomous Nonlinear Dynamical Systems

Let $\mathbf{F}:X\times\mathbb{R}^{+}\to X$ be a non-autonomous dynamical system, which is governed by $\dot{\mathbf{x}} = \mathbf{F}(\mathbf{x}, t, u)$, viz, \begin{equation} \begin{split} \dot{x}_1 ...
0
votes
0answers
39 views

Characters of Topological Group of $\mathbb{R}^n$

I am seeking to show that if $\phi :\mathbb{R}^n\rightarrow\mathbb{C}$ is a character of the topological group $\mathbb{R}^n$ then $\phi$ must have the form $\phi(x)=e^{ix\cdot\xi}$ for some $\xi$ in ...
1
vote
1answer
81 views

Uniqueness and Existence problem

I just need a bit of help with this question. If I know that $dg/dx = g^2$, and that $g(0) = g_0$, then I can solve: $$ dg/dx = g^2\\ \frac{1}{g^2} dg = dx \\ -\frac{1}{g} = x + \hat c \\ -g = ...
2
votes
2answers
465 views

Free-fall according to Newton's gravitation law

Most analysis of free-fall assume that bodies fall with constant acceleration. If however one analyses free-fall according to Newton's gravitation law, one is lead to a differential equation which I ...