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

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8
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1answer
228 views

When do Harmonic polynomials constitute the kernel of a differential operator?

Let $f$ be a real polynomial of two variables. Let $\partial_f=f\left(\frac{\partial}{\partial x},\frac{\partial}{\partial y}\right)$. Let $H$ denote the space of harmonic polynomials, i.e., ...
7
votes
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 g_2p_1^2\left(\cos(2\tau)-...
5
votes
2answers
347 views

Proving Nonhomogeneous ODE is Bounded

I am trying to prove the following: Let $x(t)$ be a solution of the IVP $$ \dot x=A(t)x+h(t), $$ where $A(t), h(t)$ continuous on $1\le t<\infty$. Further assume that $$ \int_1^\infty \| A(t)\|\,...
5
votes
2answers
4k views

Solving an initial value ODE problem using fourier transform

I am a physics undergrad and studying some transform methods. The question is as follows: $y^{\prime \prime} - 2 y^{\prime}+y=\cos{x}\,\,\,\,y(0)=y^{\prime}(0)=0\,\,\, x>0$ I am having some ...
4
votes
1answer
220 views

Formal proof of Lyapunov stability

I was trying to solve the question of AeT. on the (local) Lyapunov stability of the origin (non-hyperbolic equilibrium) for the dynamical system $$\dot{x}=-4y+x^2\\\dot{y}=4x+y^2$$ The streamplot ...
7
votes
1answer
178 views

solution of $y' = \exp \left(-\frac yx\right) + \frac yx$

Could you help me to solve equation $$y' = \exp \left(-\frac yx\right) + \frac yx;\quad y(e) = 0$$ I know how to solve 1st order linear de like $y' = \exp \bigl(-\frac 1x\bigr) + \frac yx$ but here ...
5
votes
1answer
2k views

Looking for a logically coherent book for the self-study of differential equations

I'm looking for a logically coherent book for the self-study of differential equations. Let me clarify. By logically coherent, I don't mean proofs of the limit laws, uniqueness theorems etc. By ...
4
votes
1answer
7k views

Polar coordinates differential equation

I have the following ODE: $$\dot x=-y(x^2+y^2), \dot y=x(x^2+y^2)$$ I want to sketch the phase portrait (manually) and I want to find the flow $\phi_t$, the orbit $O(x_0)$ and the limit set $\omega(...
3
votes
1answer
2k views

system of ode with non-constant coefficient matrix

I am sorry but I haven't learn any method to solve this kind of problem if the given matrix is non-constant. $$\begin{pmatrix}x\\y\end{pmatrix}^\prime=\begin{pmatrix} 1&-\cos t \\ \cos t & 1\...
3
votes
1answer
137 views

Lipschitz continuity and differential equations

Does anyone have any ideas for this one? could use some help.
2
votes
1answer
1k views

Prove Friedrichs' inequality

I'm trying to show that the theorem (Friedrichs' inequality) in my book: Assume that $\Omega$ be a bounded domain of Euclidean space $\Bbb R^n$. Suppose that $u: \Omega \to \Bbb R$ lies in the ...
2
votes
3answers
703 views

How does an integrating factor geometrically “uncurl” a vector field?

We know that certain 1-D forms $m(x,y,z)\,dx + n(x,y,z)\,dy + p(x,y,z)\,dz$ admit integrating factors as we teach in basic differential equations. How does the integrating factor geometrically turn ...
1
vote
0answers
186 views

Trapping region for Nonlinear ODE system?

I need to find a trapping region for $u'=-u+vu^2$ $ v'=b-vu^2$ with $b>0$. I don't know what theory to use or in wich book I can find some examples to find optimal trapping regions. Thank you ...
1
vote
1answer
3k 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 Strang'...
11
votes
8answers
6k views

how do you solve $y''+2y'-3y=0$?

I want to solve this equation: $y''+2y'-3y=0$ I did this: $y' = z$ $y'' = z\dfrac{dz}{dy}$ $z\dfrac{dz}{dy}+2z-3y=0$ $zdz+2zdy-3ydy=0$ $zdz=(3y-2z)dy$ $z=3y-2z$ ...
8
votes
2answers
195 views

Geometric series of an operator

In solving a first order linear differential equation $(1-D)y=x^2$ where $D\equiv \frac{d}{dx}$ the way I learnt was that we proceed as $y=\frac{1}{1-D}x^2=(1-D)^{-1}x^2=(1+D+D^2+D^3+\cdots)x^2=x^2+2x+...
8
votes
3answers
2k views

To what extent can you manipulate differentials like dy and dt like actual values?

I have been thinking about the differentials that we use in derivatives and integrals. For example, I have an equation: $${\int{w}{dr}} = \text{other stuff}$$ The context for this strange equation ...
8
votes
2answers
429 views

Deriving the addition formula of $\sin u$ from a total differential equation

How do we derive the addition formula of $\sin u$ from the following equation? $$\frac{dx}{\sqrt{1 - x^2}} + \frac{dy}{\sqrt{1 - y^2}} = 0$$ Motivation Let $u = \int_{0}^{x}\frac{dt}{\sqrt{1 - t^2}}...
8
votes
1answer
438 views

To get addition formula of $\tan (x)$ via analytic methods

Assume that we only know $\tan (0)=0$ and also given the relation $\tan'(x)=1+\tan^2(x)$ about $\tan (x)$ and we do not know other $\tan (x)$ relations of trigonometry. How can I get the additon ...
8
votes
1answer
209 views

Solving the differential equation $\frac{dy}{dx}=\frac{3x+4y+7}{x-2y-11}$

How do we solve the differential equation $$\frac{dy}{dx}=\frac{3x+4y+7}{x-2y-11}$$? I tried substituting $v=yx$ but I do not seem to be getting anywhere.Putting $u=x-2y$ yielded nothing better. ...
5
votes
1answer
174 views

Solutions of autonomous ODEs are monotonic

Problem. Let $I,J$ be open intervals, $\,f:I\to \mathbb R$, continuous, $\,\varphi :J\to \mathbb R$, continuously differentiable, with $\varphi[J]\subset I$, and $\varphi$ satisfying $$ \varphi'(t)=f\...
4
votes
3answers
144 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 ...
4
votes
1answer
144 views

Under which conditions a solution of an ODE is analytic function?

If I'm not wrong there is a theorem that says that if the conditions for Picard's theorem are satisfied, for an ode $\dot x=f(x,t)$, then the solution of the ode is as smooth as $f$. I think I'm not ...
4
votes
3answers
156 views

Minimizing a functional with a free boundary condition

Find the extremals of the functional $$\text{J}(y)= y^2(1) + \int_0^1 y'^2(x)dx , \ \ y(0)=1.$$ Answer: $y(x)=1-\frac{1}{2}x$ My solution: $ F (x,y,y')=y'^2(x)$ After solving the Euler ...
4
votes
1answer
246 views

How to solve this recurrence Relation - Varying Coefficient

Sir,I have two questions related to this recurrence relation. It has been messing with me for long. Because of this I couldn't proceed my work for some time .This contains a polynomial term n+2 in ...
4
votes
1answer
153 views

What does a standalone $dx$ mean?

Some literature uses $dx$, in the context of differential equations, in a confusing way without defining what it really stands for: $Mdx + Ndy = 0$ Does it mean one of the following or something ...
4
votes
3answers
2k views

Simple Harmonic Oscillator Solution

In Physics, the Simple Harmonic Oscillator is represented by the equation $d^2x/dt^2=-\omega^2x$ . By using the characteristic polynomial, you get solutions of the form $x(t)=Ae^{i\omega t} + Be^{-i\...
4
votes
4answers
429 views

Differential equation with a constant in it

Solve $$y'' + s^2y = b \cos sx$$ where $s$ and $b$ are constants. I have tried undetermined coefficients, but it makes such a mess that I keep getting lost, I also tried variation of ...
3
votes
1answer
893 views

General solution of a system of linear differential equations with multiple generalized eigenvectors

I am looking for general solutions for the linear sODE's $$\textbf{x}'(t) = A\textbf{x}(t)$$ with $t \geq 0$ and $A \in \mathbb{R}^{n \times n}$ Let focus on just real eigenvalues and eigenvectors. ...
2
votes
1answer
131 views

Exponential of matrices and bounded operators

Let $A$ be a complex $n \times n$ matrix, such that the function $t\mapsto e^{tA}x$ is bounded on $\mathbb{R}$ and nonzero, for some vector $x\in \mathbb{C}$. How can we prove that $\inf_{t\in \mathbb{...
2
votes
1answer
56 views

Analytic solution of: ${u}''+\frac{1}{x}{u}'=-\delta e^{u}$

I am trying to find the analytic solution of $${u}''+\frac{1}{x}{u}'=-\delta e^{u}$$ given the homogeneous mixed boundary conditions $${u'(0)}=0$$ $$u(1)=0$$ How would one attack such a problem? I ...
2
votes
1answer
124 views

Finding a specific improper integral on a solution path to a 2 dimensional system of ODEs

In my study of dynamical systems I was recently met with this system of ODEs: $ \dot{x}=\frac{\sinh{(y)}}{\cosh{(y)}+A\cos{(x)}} $ $ \dot{y}=\frac{A\sin{(x)}}{\cosh{(y)}+A\cos{(x)}} $ for a ...
2
votes
2answers
1k 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 ...
2
votes
1answer
31 views

Phase plots of solutions for repeated eigenvalues

I have a question with respect to phase plots of repeated eigenvalue cases. For instance suppose that one is given a matrix with the following: $$\overrightarrow{y'} = \begin{pmatrix} 3 & -4 \\ ...
1
vote
2answers
113 views

Does $e^{tx} + x = t - 1$ yield a solution/s to $\frac{dx}{dt} = \frac{e^{-tx} - x}{e^{-tx} + t}$?

Problem: Use Existence Theorem to determine if $x(t)$ implicitly defined by $$e^{tx(t)} + x(t) = t - 1 \tag{*}$$ yield a solution/s to $$\frac{dx}{dt} = \frac{e^{-tx} - x}{e^{-tx} + t} \...
1
vote
2answers
2k views

solution of first order differential equation and maximal interval

Find the solution of $x' = x^2t$ with initial value $x(0) = x_{0}$. Determine the maximal interval where it exists, depending on $x_{0}$ Please help me find the maximal interval!
1
vote
2answers
79 views

solution of a ODE with a funtion of $\dot{x}$

I have the equation: $$m\ddot{x}(t)+kx(t)=A$$ with m, k as constants and $$A = \left\{ \begin{array}{lr} a & : \dot{x}(t) <0\\ -a & : \dot{x}(t) >0 \end{array} ...
0
votes
1answer
294 views

Solve the given differential equation by using Green's function method

I am really struggling with the concept and handling of the green's function. I have to solve the given differential equation using Green's function method $\frac{d^{2}y}{dx^{2}}+k^{2}y=\delta (x-x');...
13
votes
4answers
598 views

Find a continuous function $f$ that satisfies…

Find a continuous function $f$ that satisfies $$ f(x) = 1 + \frac{1}{x}\int_1^x f(t) \ dt $$ Note: I tried differentiating with respect to $x$ to get an ODE but you get one that contains integrals - ...
9
votes
4answers
387 views

Differential Equation Math Puzzle

Dog race: Edit 2: I posted a possible answer below. However, I am unsure how the authors arrived at the solution. Maybe someone can offer an explanation. Four dogs are positioned at the corners of ...
9
votes
3answers
3k views

Solution of $y''+xy=0$

The differential equation $y''+xy=0$ is given. Find the solution of the differential equation, using the power series method. That's what I have tried: We are looking for a solution of the form $y(...
7
votes
1answer
1k 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
votes
2answers
4k 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.
5
votes
2answers
269 views

How to prove $(x^2-1) \frac{d}{dx}(x \frac{dE(x)}{dx})=xE(x)$

$$E(x)=\int_0^{\frac{\pi}{2}} \sqrt{1-x^2 \sin^2 t}\, dt$$ Where $E(x)$ is complete elliptic integral of the second kind. $u=\sin t$ $$E(x)=\int_0^{1} \frac{\sqrt{1-x^2 u^2}}{\sqrt{1-u^2}}\, du$$ $$...
5
votes
2answers
574 views

Regarding Ladder Operators and Quantum Harmonic Oscillators

When dealing with the Quantum Harmonic Oscillator Operator $H=-\frac{d^{2}}{dx^{2}}+x^{2}$, there is the approach of using the Ladder Operator: Suppose that are two operators $L^{+}$ and $L^{-}$ and ...
4
votes
1answer
207 views

Solution of $ f \circ f=f'$

Let $f:\mathbb R \to \mathbb R $ be a function such that $f \circ f=f'$ and $f(0)=0$ , I proved that $f$ is the null function. Can we prove that the same result holds if we change $f \circ f=f'$ by $...
4
votes
3answers
259 views

Prove that if $\phi'(x) = \phi(x)$ and $\phi(0)=0$, then $\phi(x)\equiv 0$. Use this to prove the identity $e^{a+b} = e^a e^b$.

I am given the following. hint Consider $f(x)=e^{-x} \phi(x)$. I am unsure how to approach this problem.
4
votes
1answer
207 views

How do you solve this differential equation using variation of parameters?

$\color{green}{question}$: How do you solve this differential equation using variation of parameters? $$y"-\frac{2x}{x^2+1}y'+\frac{2}{x^2+1}y=6(x^2+1)$$ $\color{green}{I~tried}$ . . . $using~...
4
votes
1answer
72 views

Find minimizer of the functional $l(u)= \int_{-1} ^1 u(t) \mathbb d t$

Find minimizer of the functional $ l(u)= \int \limits _{-1} ^1 u(t) \mathbb d t $ with $u(-1)=u(1)=0 $ subject to $g(u)=\int \limits _{-1} ^1 \sqrt{1+u'(t)} \mathbb d t=π $. I solved it using ...
4
votes
1answer
380 views

Using the Lambert W to express a solution of a differential equation.

I solved a differential equation some time ago and I need to solve for $y$. How can we solve for $y$ using the Lambert W function? $$C_1+x = e^y+Cy$$