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

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11
votes
2answers
14k views

Explanation and Proof of the fourth order Runge-Kutta method

Runge-Kutte 4th order method is a numerical technique used to solve ordinary differential equation of the form $dy/dx=f(x,y), y(0)=y_0$ It gives $y_{i+1}$ in the form $y_{i+1} = ...
21
votes
2answers
399 views

Periodic orbits of “even” perturbations of the differential system $x'=-y$, $y'=x$

Fix some even functions $f$ and $g$, differentiable, such that $f(0)=g(0)=0$ and $f'(0)=g'(0)=0$, and consider the autonomous differential system $$\left\{\ \begin{array}{lcr}x'&=&-y+f(x)\\ ...
13
votes
4answers
4k views

Can this gravitational field differential equation be solved, or does it not show what I intended?

This is the equation I'm having trouble with: $$G \frac{M m}{r^2} = m \frac{d^2 r}{dt^2}$$ That's the non-vector form of the universal law of gravitation on the left and Newton's second law of ...
13
votes
3answers
519 views

Fourth Order Nonlinear ODE

I was looking at an ode $w^{(4)} + w^3 = 0$ with initial conditions $[w'''(0),w''(0),w'(0),w(0)]=[1,0,0,0]$. I can see via maple that there is a blowup around 3.7. I was wondering if there was a way ...
10
votes
4answers
10k 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 ...
10
votes
2answers
2k views

Euler-Lagrange, Gradient Descent, Heat Equation and Image Denoising

For an image denoising problem, the author has a functional $E$ defined $$E(u) = \iint_\Omega F \;\mathrm d\Omega$$ which he wants to minimize. $F$ is defined as $$F = \|\nabla u \|^2 = u_x^2 + ...
9
votes
1answer
10k views

General Solution of a Differential Equation using Green's Function

My father recently lent me an old textbook of his, called Mathematical Methods of Physics by Mathews and Walker. I am working on the following exercise. Consider the differential equation ...
9
votes
2answers
1k views

Particular solution to a Riccati equation $y' = 1 + 2y + xy^2$

The equation is $y' = 1 + 2y + xy^2$. I've tried $mx+n$, $ax^m$, even $\tan x$ as candidates for particular solution where $a,m,n \in \mathbb Q$, but it did not work. Can anyone find one particular ...
7
votes
3answers
323 views

Why it is absolutely mistaken to cancel out differentials?

In many of my physics courses, (don't worry, this is a mathematics question!) My teachers cancel out differentials, and every time, they say: "If a mathematician saw me canceling out this ...
11
votes
3answers
18k views

Definition of a Differential Equation?

Here is one definition of a differential equation: "An equation containing the derivatives of one or more dependent variables, with respect to one of more independent variables, is said to be a ...
7
votes
1answer
287 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 $W$.
12
votes
1answer
844 views

Recursive solutions to linear ODE.

When finding the solutions to the simple ODE $$ y'- mxy= x^n \text{ ; } y(0) = 0$$ I found the following: Let $P_n$ be the particular solution for each integer exponent $n$. Then if we define ...
8
votes
1answer
223 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., ...
5
votes
2answers
341 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 \| ...
5
votes
2answers
383 views

The Green’s function of the boundary value problem

What is the Green’s function of the boundary value problem $$ \frac{\mathrm d^2 y}{\mathrm d x^2}-\frac{1}{x}\frac{\mathrm dy}{\mathrm dx}=1,\quad y(0)=y(1)=0, $$ this boundary problem is not self ...
7
votes
1answer
177 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
3answers
6k views

Fourier Series for $|\cos(x)|$

I'm having trouble figuring out the Fourier series of $|\cos(x)|$ from $-\pi$ to $\pi$. I understand its an even function, so all the $b_n$s are $0$ $$a_0 = \frac 2 \pi \int_0^\pi |\cos(x)|\,dx = ...
4
votes
1answer
198 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 ...
4
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 ...
3
votes
1answer
133 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
683 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
168 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
2answers
491 views

Comparison theorem for ODE

Here is something I'm trying to prove: Conjecture: Suppose $f'(x) \leq \phi(f(x), x)$ and $f(a)=\alpha$. Suppose $g'(x)=\phi(g(x),x)$ and $g(a)\geq \alpha$. Then $f(x)\leq g(x)\,\,\forall x$. ...
1
vote
1answer
105 views

Basic Reference material about ODEs such as saparability with calculations and examples?

I am trying to show this kind of non-linear $y''''=y'y''/(1+x)$ in normal form. For example here if $y=e^{x}\rightarrow y^{(n)}=e^{x}\rightarrow x=-1$, where $y^{(n)}$ ...
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: $y' = z$ $y'' = z\dfrac{dz}{dy}$ $z\dfrac{dz}{dy}+2z-3y=0$ $zdz+2zdy-3ydy=0$ $zdz=(3y-2z)dy$ $z=3y-2z$ ...
9
votes
1answer
299 views

Solution of differential equation related to Normal density

Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, $$\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}{2}}, \forall x\in\mathbb{R}.$$ Given $0<\sigma\le 1$. I wish to know whether there ...
8
votes
1answer
429 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
2answers
421 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 - ...
8
votes
3answers
1k 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 ...
7
votes
1answer
200 views

prove that the following function is: $f(x) = 0$

let $f: [0,1] \to \mathbb R$ , $f$ is differentiable $f(0) = 0$ $|f'(x)|\le|f(x)|$ for $x\in [0,1]$ prove that $f(x) =0$ for $x\in [0,1]$ i believe that i need to somehow use the ...
4
votes
1answer
146 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
143 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
140 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
4answers
424 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 ...
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} + ...
3
votes
2answers
66 views

How can I find a solution of second order ODE with variable coefficients?

I want to find a solution of $$ \left(\frac{d^2}{dx^2} + (1+x^2)^{-1/2} \frac{d}{dx} + c \right)f(x) = 0 $$ where $x \in \mathbb R$ and $c$ is a real constant.
3
votes
1answer
808 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
55 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
122 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 ...
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 ...
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} ...
1
vote
1answer
78 views

number of points of tangency of the zero divergence vector field and the equator of the sphere.

Let $V$ be vector field on the sphere $S^2$ and $\operatorname{div} V=0$. What is the minimum number tangency points of this vector field and the equator of the sphere?
1
vote
1answer
102 views

Does there exist a system which has closed orbit but no constant one?

Can you please give me an example of an ODE system which has no constant orbit or fixed point but it has closed orbit? Thank you very much
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!
0
votes
1answer
274 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 ...
13
votes
4answers
551 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
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 ...
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.