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

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370 views

Prove that a map is continuous

Let $r:[0,1]\to\mathbb R$ be a continuous function and let $u_\lambda$ be the unique solution of the Cauchy Problem: $$\begin{cases}u''(t)+\lambda r(t)u(t)=0,\quad\forall t\in [0,1],\\ ...
7
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1answer
32 views

Initial value problem

Solve the following initial value problem: $$\frac{d^2y}{dt^2}+2\frac{dy}{dt}+5y=0; y(0)=0 \text{ and } y'(0)=2 $$ I started off with the characteristic equation which is: $$ r^2+2r+5=0 $$ Using ...
7
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3answers
209 views

Distribution theory and differential equations.

How does distribution theory plays role in solving differential equations? This question might seem to be very general. I will try to explain, please bear with me. I understand, distributions make it ...
7
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1answer
166 views

$\nabla \cdot f + w \cdot f = 0$

Let $w(x,y,z)$ be a fixed vector field on $\mathbb{R}^3$. What are the solutions of the equation $$ \nabla \cdot f + w \cdot f = 0 \, ? $$ Note that if $w = \nabla \phi $, then the above equation is ...
7
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3answers
229 views

Differential equations solveable independently of coordinate system?

Looking from a physics viewpoint ODEs tend to look very differently when setting up the problem in different coordinate systems. For instance the Laplacian in spherical coordinates involves way more ...
7
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4answers
349 views

A Handwaving Proof of a Specific Existence and Uniqueness Theorem

My problem is as follows: Given the second order homogeneous linear differential equation with constant coefficients $$a\frac{d^2y}{dx^2}+b\frac{dy}{dx}+c\,y(x)=0,$$ is there a good heuristic ...
7
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3answers
530 views

How to find the general solution of $xy''-(2x+1)y'+x^2y=0$ when we know the general solution of $y''+2y'+xy=0$?

Given that the general solution of $y''+2y'+xy=0$ is $y=C_1\int_0^\infty ...
7
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1answer
384 views

ODE Laplace Transforms: what impulse brings an oscillating system to rest?

$2y''+y'+2y=\delta(t-5)$ $y(0)=0, y'(0)=0$ Consider the system given by ODE above in which an oscillation is excited by a unit impulse at $t=5$. Suppose that it is desired to bring the system to ...
7
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1answer
94 views

Solve $y' + \frac1y + \frac1x =0$ Differential Equation

Do you have any suggestions for how to sole this differential equation? $y'+\frac1x + \frac1y =0$ ? :) I tried solving this by changing variable in the form of $v=x^\alpha*y^\beta$ but it didn't ...
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2answers
50 views

Show that any solution of second order differential equation has atmost a countable number of zeroes $?$

Question : Considered the second order differential equation $y''(t) + a(t) y'(t) + b(t) y(t) = 0$. then any solution of second order differential equation has atmost a countable number of zeroes ...
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2answers
85 views

Why is Existence and Uniqueness for Navier-Stokes Easier in 2-D than in 3-D?

I know that existence and uniqueness for incompressible viscous flow in the 2-D case has already been established$^1$, and that doing the same for the 3-D case has yet to be shown. Not only that, but ...
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1answer
176 views

how to solve this differential equation $\frac{dy}{dx} = \frac{1+xy}{x(1-xy)}$ by substitution?

I've tried with this differential equation $\displaystyle \frac{dy}{dx} = \frac{1+xy}{x(1-xy)}$ , put $u=xy$ then $\displaystyle\frac{du}{dx}=x\frac{dy}{dx}+y$ So, It will be after editing ...
7
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1answer
111 views

dropping a particle into a vector field, part 2

Okay, so earlier I posted this question "dropping a particle into a vector field " as sort of a feeler question as i study line integrals in order to go into surface integrals and eventually ...
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1answer
370 views

$\frac{dS}{d\rho}$ Factor arising

To get details see: equations 29,30,31,34,44,50,51 We have known some solitary wave solutions, given by(equations 1 to 5) $$ \phi_1=p_1\cos \tau \tag{1}$$ $$\phi_2=\frac16 ...
7
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1answer
2k views

A Differential Equation with Trigonometric Coefficients

Suppose we have the following second-order differential equation: $\cos^2(x)y'' -\sin(x)y' + y = 0$ How do we determine its general solution? I couldn't even guess a particular solution; all my ...
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2answers
330 views

Deriving the addition formula for the lemniscate functions from a total differential equation

The lemniscate of Bernoulli $C$ is a plane curve defined as follows. Let $a > 0$ be a real number. Let $F_1 = (a, 0)$ and $F_2 = (-a, 0)$ be two points of $\mathbb{R}^2$. Let $C = \{P \in ...
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1answer
206 views

$\int_3^5{\frac{x^2}{1+x^2}dx}$ by differentiation under the integral

I'm trying an easy problem to get my bearings using the method here. The integral is $$\int_3^5{\frac{x^2}{1+x^2}dx}$$. I would like to proceed, if possible to solve by defining: $$F(y) = ...
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3answers
303 views

Symmetry of Solution to Classical 3-Dimensional Isoperimetric Problem

A while ago I attempted to solve the classical isoperimetric problem in 3-dimensions, namely "Find the surface that has the smallest surface area for a given volume". At that time for me to write ...
7
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1answer
100 views

$\cos(x+y)\frac{\partial z}{\partial x}+\sin(x+y)\frac{\partial z}{\partial y}=z+\frac{1}{z}$

Having difficulty in solving the following partial differential equation: $$\cos(x+y)\frac{\partial z}{\partial x}+\sin(x+y)\frac{\partial z}{\partial y}=z+\frac{1}{z}.$$ Will it be easier if we ...
7
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1answer
132 views

Limit points of the differential system $\dot {x}=y-x+x^3$, $\dot{y}=-x$

Consider the following system of differential equations: $$\dot {x}=y-x+x^3,\qquad \dot{y}=-x.$$ By linearization, it's easy to see that $(0,0)$ is a (nonlinear) sink. Show that there exists an ...
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1answer
73 views

global behavior of an ODE system

Consider the ODE system $$ (x',y')=f(x,y) $$ where $$ f(x,y)=((1-x/2-y/2)x,(-1/4+x/2)y) $$ in the first (open) quadrant. It is not hard to show that $z_0=(1/2,3/2)$, which is a equilibrium of the ...
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1answer
184 views

Solutions for $ \frac{dy}{dx}=y $?

Al-right, this may be a very basic question but I'm confused about this. We all know that one differential equation can only have one solution. Consider: $$ \frac{dy}{dx}=y $$ The solution is: $$ ...
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1answer
250 views

First-term approximation for singular perturbation of ODE (with two turning points)

I'm reading "Introduction to Perturbation Methods" by Mark Holmes, and I came across an exercise that I don't know how to approach. As I decided to independently read this book, I have no ...
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214 views

Riccati differential equation

I have to solve the following equation : $$ \frac{dx}{dt}(t)=-q x^2(t) +1 $$ with $x(0)=1$ and $q>0$. At first I consider the two cases: $q=1$, then I take the change of variable $ x= ...
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1answer
130 views

Periodic solution of differential equation y′=f(y)

Let $f∈C^∞(ℝ^2,ℝ^2)$ and $∀x∈ℝ^2$ $f(kx)=k^2f(x)$ for $k∈ℝ$ Show that any periodic solution of $y′=f(y)$ is constant. My attempt : Let $\lambda \in \mathbb{R}$. Let $g$ a periodic solution ...
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1answer
2k views

Restricted Three-Body Problem

The movement of a spacecraft between Earth and the Moon is an example of the infamous Three Body Problem. It is said that a general analytical solution for TBP is not known because of the complexity ...
7
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1answer
396 views

Grothendieck connections and jets

The following question is based on some remarks in section I.2 of Deligne's book Equations Différentielles à Points Singuliers Réguliers. Let $X$ be a smooth complex variety and ...
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1answer
71 views

Solve second order differential equations

There are two differential equations that I could not solve. Can someone please help me solve them? $$ (x^2+y^2)y′′-y(y^{′})^3+xy′-y=0 $$ and $$ xy^2y′′+2y^2y′-4xy(y^{′})^2+2x^2(y^{′})^3=0, $$ ...
7
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1answer
130 views

Derivation of Schrödinger's equation

I recall a famous quote of the late physicist Richard Feynman: Where did we get that from? It's not possible to derive it from anything you know. It came out of the mind of Schrödinger. This ...
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5answers
816 views

Finding a non constant solution to $ (x')^2+x^2=9 $

How do I find a non-constant solution this equation? I've tried to solve for $x$, but the final answer should be in the form of $x(t)=...$ $ (x')^2+x^2=9 $ I'm not sure where to start.
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2answers
335 views

Determinant called Grammian

Famously, if functions $f_1,f_2,…,f_n$, each of which possesses a derivative of order $n-1$, are linearly independent on the interval $I$, if $$ \det\left( \begin{array}{ccccc} f_1 & f_2 & ...
6
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3answers
805 views

Solving $-u''(x) = \delta(x)$

A question asks us to solve the differential equation $-u''(x) = \delta(x)$ with boundary conditions $u(-2) = 0$ and $u(3) = 0$ where $\delta(x)$ is the Dirac delta function. But inside ...
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5answers
189 views

Check $\;y=\dfrac{\sin x}{x}\;$ is solution of $\;xy'+y=\cos x\;$

How can I check that $\;y=\dfrac{\sin x}{x}\;$ is a solution of $\;xy'+y=\cos x\;$?
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3answers
454 views

Common term for differential equations and recurrence relations

Recently I have been working with recurrence relations (mostly linear), and systems of coupled recurrence relations. I have noticed a lot of common ground with differential equations. In a way, you ...
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3answers
3k views

Differential Equations without Analytical Solutions

In many talks, I have heard people say that the differential equation they are interested in has no analytical solution. Do they really mean that? That is: Can you prove a differential equation ...
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3answers
421 views

Why do we chose exponential function as a trial solution for second order linear differential equation with constant coefficient?

Why do we chose exponential function as a trial solution for second order linear differential equation with constant coefficient ? Can any other function be taken as a trial solution ?
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4answers
5k views

Can someone intuitively explain what the convolution integral is?

I'm having a hard time understanding how the convolution integral works (for Laplace transforms of two functions multiplied together) and was hoping someone could clear the topic up or link to sources ...
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2answers
2k views

Can a differential equation have non unique solutions?

There are theorems of existence and uniqueness of differential equations. I was wondering if it is possible that a differential equations has a solution but it is not unique.
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3answers
3k views

Integrating with respect to different variables

I have started reading a book on differential equations and it says something like: $$\frac{dx}{x} = k \, dt$$ Integrating both sides gives $$\log x = kt + c$$ How is it that I can ...
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3answers
114 views

Why don't the roots of this characteristic equation correspond to the given solution of this 2nd order ODE?

I am asked to solve $$ y'' + 9y = 6\mathrm{sin}(3x) $$ using the method of undetermined coefficients. The characteristic equation of this 2nd order ODE is $$ \lambda^2 + 9\lambda = 0 $$ and its ...
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3answers
253 views

What are other solutions to this differential equation, “similar” to $\sin x$ and $e^x$?

I've been studying electronics, where they make great use of the relationship between the sine and exponential functions ($e^{i \omega t} = \cos{\omega t} + i \sin \omega t)$. This relationship is ...
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3answers
336 views

two identical point charges can't collide

I've convinced myself intuitively that if you place two massless classical particles with the same charge in $\mathbb{R}^n$, with arbitrary initial velocities and (distinct) positions, they will never ...
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1answer
915 views

How can I show that $y'=\sqrt{|y|}$ has infinitely many solutions?

Show that the first order differential equation $y'(x)=\sqrt{|y(x)|}$ with intial value $y(1/2)= 1/16$ has infinitely many solutions on the interval [−1, 1]. My thought were to show that this ...
6
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2answers
197 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 ...
6
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3answers
730 views

Square root of differential operator

If $D_x$ is the differential operator. eg. $D_x x^3=3 x^2$. How can I find out what the operator $Q_x=(1+(k D_x)^2)^{(-1/2)}$ does to a (differentiable) function $f(x)$? ($k$ is a real number) For ...
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5answers
1k views

Modelling forces acting on a sail

I'd like to create a basic model of the forces acting on a sail (wind sail, like a tail ship) A couple of things I was thinking about: 1) can create a very simple model where wind is 'one' force ...
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4answers
579 views

Derivative of square of derivative?

I was trying to solve this differential equation: $$2yy'' + 3y'^2 = 4y^2 $$ And I found this way to solver it: http://eqworld.ipmnet.ru/en/solutions/ode/ode0344.pdf but I don't understand why $w'_y ...
6
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3answers
228 views

Solutions of homogeneous linear differential equations are a special case of structure theorem for f.g. modules over a PID

In this other post, Qiaochu Yuan comments that the solutions for the homegeneous linear differential equation with constant coefficients are a special case of the structure theorem for finitely ...
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1answer
186 views

Finite dimensional spaces

What are the finite-dimensional spaces $W$ of differentiable functions with this property: If $f$ is in $W$, then $\frac{df}{dx}$ is in $W$.
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204 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 ...