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

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9
votes
3answers
291 views

Nicer expression for the following differential operator

I have the following sequence of differential operators: $$D_n = \underbrace{t \partial_t t \partial_t \dots t \partial_t}_{\text{$n$ times}}.$$ Is there any expression involving a sum of "normal" ...
3
votes
3answers
347 views

Differential equation: autonomous system

This isn't homework. I have no idea what theorems I should be looking at to solve this. Guidance, partial and total solutions are all welcomed. Let $f$ be a locally lipschitz function in an open ...
3
votes
2answers
141 views

Find a general control and then show that this could have been achieved at x2

Determine the general form of $u_0, u_1 ~\text{and} ~ u_2$ if a system of difference equations of the form $$x_{n+1} = Ax_n + Bu_n,$$ where: $$A = \begin{pmatrix} 3 & 2 & 2 \\ -1 ...
3
votes
2answers
336 views

Generating unitary matrices numerically - “close” to the identity element

EDIT: broke this into two parts - for these were two different questions. For numerically obtaining the stabilities of a matricial equation, i need to generate an ensemble of matrices that are ...
2
votes
2answers
1k views

A problem For the boundary value problem, $y''+\lambda y=0$, $y(-π)=y(π)$ , $y’(-π)=y’(π)$

For the boundary value problem, $y''+\lambda y=0$ $y(-π)=y(π)$ , $y’(-π)=y’(π)$ to each eigenvalue $\lambda$, there corresponds Only one eigenfunction Two eigenfunctions Two linearly ...
2
votes
2answers
5k views

Find the differential equation of all circles of radius a

Can someone please post a detailed step-by-step procedure. Given the circle with a radius a, what is the differential equation of the circle.
1
vote
3answers
104 views

Deducing the exact solution of a ODE

In page 53 of Arieh Iserles's A first course in the numerical analysis of differential equations, he presents the following ODE: $(\vec{y})'=\Gamma\cdot\vec{y}$, $\vec{y}(0)=\vec{y_0}$ Using the ...
1
vote
2answers
372 views

How to know if a point is analytics or not?

So I have the equation y" + [(x-x^3)/x]y' +[(sinx)/x]y = 0 My x nought it equal to 0. I know this is a singular point because my denominator is equal to ...
1
vote
3answers
115 views

Solving $\frac{dx}{dz}-\frac{2x}{z}=1$

Please can someone solve this? $$\frac{dx}{dz}-\frac{2x}{z}=1$$ Please this is only part of my homework question. I am stuckwith here. Please teach me this solution thank you:)
1
vote
2answers
138 views

solving this second order ode

Consider the second order ODE where $ (k-x)^2 y''+6(k-x)y'+12y=F(x) $ where $k$ is some constant. I want to compute the real valued general solution. progress: guess $(k-x)^{m}$ to be the solution ...
1
vote
4answers
316 views

Finding all functions $f$ satisfying $f'(t)=f(t)+\int_a^bf(t)dt$

I am trying to find all functions f satisfying $f'(t)=f(t)+\int_a^bf(t)dt$. This is a problem from Spivak's Calculus and it is the chapter about Logarithms and Exponential functions. I gave up ...
0
votes
1answer
254 views

Consider Van der Pol’s equation:

Consider Van der Pol’s equation: $$y′′−0.2(1−y^2)y′+y=0,\qquad y(0)=0.1,\ y′(0)=0.1$$ (i) Find the approximate solution for this problem using the Taylor series method. Your expansion should include ...
17
votes
3answers
5k views

Differential equations and Fourier and Laplace transforms

Why do both the Fourier transform and the Laplace transform appear in the study of differential equations? I've never understood why there are some situations where the Fourier transform is used and ...
13
votes
4answers
2k 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 motion ...
9
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 + ...
10
votes
2answers
423 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 ...
7
votes
2answers
930 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 ...
10
votes
3answers
12k 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 ...
8
votes
1answer
209 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., ...
4
votes
1answer
4k views

Best Book For Differential Equations?

I know this is a subjective question, but I need some opinions on a very good book for learning differential equations. Ideally it should have a variety of problems with worked solutions and be ...
16
votes
5answers
636 views

history of the double root solution of $ay''+by'+cy=0$

Motivation: It is a well-known fact that $ay''+by'+cy=0$ has solutions which are found from substituting the ansatz $y=e^{\lambda t}$ into the DEqn. It turns out that we replace the calculus problem ...
7
votes
1answer
156 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
4k 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 ...
3
votes
1answer
111 views

Lipschitz continuity and differential equations

Does anyone have any ideas for this one? could use some help.
2
votes
3answers
506 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 ...
11
votes
8answers
4k 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
275 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
2answers
363 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
189 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 ...
6
votes
2answers
2k views

Finding integrating factor when IF will be a function of x and y

I'm not finding any resource or description or systemic methodology to find integrating factors when the integrating factor will be a function of both x and y. I'm on this problem, $$ ( y - xy^2 ) ...
4
votes
3answers
122 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
97 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
1k 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
2answers
605 views

Applied ODEs in trajectory problem

I'm having a hard time solving this problem: Let there be a town $A$ in a shore of a river. Let $x=0$ be the shore. Let $(0,0)$ be the location of the town. Let $B$ be another town, in the ...
4
votes
4answers
394 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 ...
2
votes
2answers
59 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.
2
votes
1answer
59 views

does there exist an system which has closed orbit but not constant one?

Can you please give me an example of an ODE system which has no constant orbit or fixes point but closed orbit? Thank u very much
2
votes
2answers
505 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
66 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
72 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
2answers
770 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
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 ...
0
votes
1answer
101 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 $y^{(n)}$ ...
13
votes
4answers
401 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 - ...
6
votes
2answers
178 views

Function whose inverse is also its derivative?

What are some good examples of a function $f : \mathbb{R} \to \mathbb{R}$ where its derivative is equal to its inverse? I attempted to find a monomial that satisfied it by starting with $f(x) = ax^b$ ...
5
votes
2answers
215 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. $u=\sin t$ $$E(x)=\int_0^{1} \frac{\sqrt{1-x^2 u^2}}{\sqrt{1-u^2}}\, du$$ ...
4
votes
1answer
136 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? $$y"-\frac{2x}{x^2+1}y'+\frac{2}{x^2+1}y=6(x^2+1)$$ $\color{green}{I~tried}$ . . . ...
4
votes
1answer
305 views

Sturm-Liouville Questions

In thinking about Sturm-Liouville theory a bit I see I have no actual idea what is going on. The first issue I have is that my book began with the statement that given $$L[y] = a(x)y'' + b(x)y' + ...
3
votes
1answer
91 views

Bessel Equations Addition Formula

So, I'm considering yet another tricky proof involving Bessel Functions. Basically, I'm trying to figure out how the following is true: $$J_n(\alpha + \beta) = \sum_{m = -\infty}^\infty ...