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

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Dynamic Pari-Mutuel Markets, generalizing to multiple discrete outcomes

In the paper http://dpennock.com/papers/pennock-ec-2004-dynamic-parimutuel.pdf the author describes a mathematical method creating a dynamic pari-mutuel market with an automated market maker. The ...
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2answers
27 views

Rearrange solution of differential equation involving SHM

This question involves simple harmonic motion (SHM). I am struggling to work out how to rearrange: $x(t)=A\cos(\omega t)+B\sin(\omega t)$ (Which is the solution of the differential equation ...
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0answers
12 views

Under which conditions the solution to a linear system of ODE has a limit?

Consider a system of the form; $$\mathbf{x}'(t)=A(t)\mathbf{x}(t)+\mathbf{f}(t),$$ $$\mathbf{x}(0)=\mathbf{b},$$ where $$\mathbf{x}(t)=(x_1(t),\ldots,x_n(t)),$$ ...
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0answers
15 views

local bifurcation and unfoldings of vector field

Problem: Consider the scalar differential equation $x˙ = x^3 = f^*(x)$ in a neighborhood of the fixed point $x = 0$. consider – $f_1(x,μ) := μ + x^3$ – $f_2(x,μ) := μx^3$ – $f_3(x,μ) := μ + ...
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1answer
32 views

Solving Non-homogenous System (repeated case)

I have the following system. $\vec{x^{'}}(t)=\begin{pmatrix}4&-2\\8&-4 \end{pmatrix} \vec{x}+ \begin{pmatrix}t^{-3}\\-t^{-2}\end{pmatrix}$ I get $\lambda=0$ and the eigenvector of $$4x_1=2x_2 ...
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1answer
24 views

bifurcation with more than parameter

Problem: Consider the scalar differential equation depending on the parameters $\alpha_1, \alpha_2$ ∈ $\Re$ $x˙ = \alpha_1 + \alpha_2 x − x^2$. Find a change of coordinates $y = \phi(x)$ such that ...
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1answer
16 views

Laplace transform of convolution with no function of t

Instructions: Evaluate the given Laplace transform. Do not evaluate the integral before transforming. Problem Given: $\mathscr{L}\{\int_0^t e^{-\tau} cos\tau d\tau \}$ My Problem: To treat this as ...
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1answer
15 views

Proof that laplace's equation is rotationally invariant using chain rule

Suppose $(x, y)$ and $(p, q)$ are coordinates in the plane related by rotation around a fixed point $(a, b)$, as follows: $$\begin{bmatrix} p\\ q\end{bmatrix} = \begin{bmatrix} \cos(t) & -\sin(t) ...
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2answers
17 views

Rewrite the equation as a system of equations

Can someone explain the steps to getting this solution. I am unsure how its possible to add the matrices of different sizes. $(\sin(t)){y}'''-2{y}''+{y}'+3y=e^t$ The solution is supposedly: ...
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question about a proof: sequence of picard iteration converges uniformly

Given $u'(t)=t\cdot u(t)+t^3$ with $u(0)=0$ I want to show that $$u_{n+1}(t)=u(t_{0})+\int_{t_0}^{t}f(x,u_{n}(x))dx$$ converges uniformly on $[-b;b]$ and solves the differential equation. After ...
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1answer
14 views

Sturm–Liouville Orthogonality problem

I have a differential equation $$X''(x) - 2X'(x) + \lambda X(x) = 0$$ with boundary conditions $$X(0) = X(a) = 0$$ I came up with eigenvalues of $\lambda_n = 1 + \frac{n^2 \pi ^2}{a^2}$ and ...
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3answers
36 views

Matrix norm in Banach space

How can I calculate the following matrix norm in a Banach Space: $$ A=\begin{pmatrix} 5 & -2 \\ 1 & -1 \\ \end{pmatrix} ?$$ I have tried ...
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1answer
19 views

Solving this linear second order Cauchy problem

Here is my problem, I know that a function is a solution of this linear Cauchy problem $$ \left\{ \begin{array}{rcl} y'' &=& \frac{x^2+6}{4}y,\\ y(0)&=&0,\\ y'(0)&=& ...
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2answers
40 views

What is the answer to $\int x(t)dt$?

$\int x(t)dt$? I'm trying to solve a differential equation, but I've hit a strange brick wall that I never used to have a problem climbing over. This question is about mechanics & the equation ...
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2answers
41 views

ODE using Weierstrass's P function

I need a hint for the following problem. "Solve $(x')^2=x^3 − 3x^2 − 4x + 12$ with the initial with initial condition $x(0)=3$". I know I should somehow use Weierstrass's $P$ function because it ...
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1answer
22 views

Differential equation using Euler's method

$y'(t)=y(t)^2$ How can this be solved using Euler's method (preferably) or another diff eq technique?
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2answers
63 views

Finding a particular solution to the non-homogenous system

I have the following problem $\vec{x}^{'}(t)=\begin{pmatrix} 2 & -5\\1 & -2 \end{pmatrix}\vec{x} + \begin{pmatrix} \csc t\\ \sec t \end{pmatrix}$ Step 1) Find the Eigenvalues ...
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1answer
30 views

PDE from London's Equation with Cylindrical Symmetry

The question is from ISSP by Kittel and as follows: (a)Find a solution of the London equation that has cylindrical symmetry and applies outside a line core. In cylindrical polar coordinates, we want ...
3
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2answers
43 views

$y''+2y'+5y=0$, initial value problem with Laplace transform?

here is the question: $$ {\rm y}''\left(t\right) + 2\,{\rm y}'\left(t\right) + 5\,{\rm y}\left(t\right) = 0; \qquad\qquad {\rm y}\left(0\right) = 2\,,\quad {\rm y}'\left(0\right) = -1. $$ ...
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2answers
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Lifting Trial Functions for Second Order ODES

I have a general question here, I've been doing non-homogenous second order differential equations. As you know, sometimes when finding the particular integral, the general trial function already ...
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0answers
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Wave Characteristic diagram question

Say $F(x)=v(x)=0$ when $|x| \geq a$ whilst $F(x)=h(x)$ and $v(x)=c \ h'(x)$, both when $|x|<a$. As we know, the wave equation in D'Alembert's form is: $$ ...
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2answers
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Inverse of matrix with varying parameters

Ok so I need some sort of verification on this. I have run into this matrix $\begin{pmatrix} e^t&3e^{-t}\\e^t&e^{-t} \end{pmatrix}$ and I need to find the inverse of this matrix. The book ...
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1answer
30 views

Show the equality holds for any $x \in [0, \pi]$

We are considering a $2\pi$ periodic function defined on $x\in(-\pi,\pi)$ by $$f(x) = \pi - x, 0<x<\pi $$ and 0 otherwise. I already computed the full Fourier series is equal to: $$f(x) = ...
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0answers
20 views

Heaviside function and solutions

I'm currently trying to solve a second ODE with a heavy side function for Homework (so cant post question) and to find the constants of integration use continuity of y and y' and was wondering are ...
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2answers
28 views

where does the modulus go when cancelling $e$ and $\ln$ in this problem?

So I did this problem today: Show that $\frac{dy}{dx} = yx^2$ can be written as $y = Ae^{\frac{x^3}{3}}$ my solution is shown below: $$ \frac{dy}{dx} = yx^2 $$ $$ \frac{1}{y} dy = x^2 dx $$ $$ ...
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0answers
27 views

Decoupling system of two partial differential equations

If I have the following systems of PDE $$ u_t+x^2u_{xx}-\dfrac{h_1(t)}{h_0(t)}e^{-(v-u)}-\dfrac{h_0'(t)}{h_0(t)} = 0,\\ v_t-\dfrac{h_0(t)}{h_1(t)}e^{-(u-v)}-\dfrac{h_1'(t)}{h_1(t)} = 0, $$ where ...
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0answers
16 views

non-dimensional equation [on hold]

Consider non-dimensional equation for the height at the highest point is given by \begin{equation} h(\mu)= \frac{1}{\mu}- \frac{1}{\mu^2} \log_e(1+\mu) \end{equation} $0<\mu\ll 1.$ Determine to ...
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1answer
59 views

Initial Value Problem for $y''+xy'+x^2y=0$

Does anyone have a solution for the initial value problem: $$y''+xy'+x^2y=0, y(0)=1, y'(0)=1 ?$$ I try power series solution but I have trouble finding a pattern for the general term of the series.
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Find the Laplace transform of: [on hold]

$$h_3(t)e^t$$ Here $3$ is the subscript of $h(t)$ used to denote the switch property of the transform.
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22 views

Using Fourier Series to solve differential equations [on hold]

Given the equation: $\frac{d^{2}i}{dt^{2}} + 10\frac{di}{dt} + 10i = \frac{dv}{dt}$ in which $v(t) = 10(\pi ^{2} - t^{2})$ and $-\pi \leq t\leq \pi $, $v(t+2\pi ) = v(t)$. Find a particular solution ...
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1answer
15 views

Find Laplace Transform of the following function

How do I find the Laplace transform for the function: $f(t)=t, 0 \leq t \leq 1$ and $2-t, t \geq 1$ I tried looking up the process online, but it remains unclear to me. Thanks in advance!
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1answer
36 views

Inverse Function Differential Equation [duplicate]

For the differential equation $$\frac{d}{dx}[y(x)]=y^{(-1)}(x)$$ where $y^{(-1)}(x)$ is the inverse of $y(x)$, find y(x). I gave up on finding the solution analytically pretty quickly and decided ...
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0answers
27 views

Closed form for a sequence defined recursively

Let $a_k$ be a sequence such that $a_0=0, a_1=0, a_2=1, a_3=1$ and $$a_{k+4}=-\frac{a_{k}+ka_{k+2}}{(k+1)(k+2)}$$ for $k\ge 0$. My question is: Is a closed form formula for $a_n, n\ge 4$ possible? ...
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1answer
19 views

Modeling with Differential Equations - Help?!?!

So here's the problem that I'm working on at the moment: Tank 1 initially contains 50 gals of water with 10 oz of salt in it, while Tank 2 initially contains 20 gals of water with 15 oz of salt in ...
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If $f(x) = Ax$, show that for all $t \in \mathbb{R}$, the extreme $x_n = x_n(t)$ of polygon converges to $e^{At} x_0$.

Let $f$ is a vector field in $\mathbb{R}^n$, $x_0 \in \mathbb{R}^n$ and $x_{k+1} = x_k + f(x_k)\Delta t$, $k= 0,1,...,n-1$, where $\Delta t = \frac{t}{n}$. A polygon whose points are the $x_i$ ...
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0answers
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How to plot $\dot{x}= Ax + Bu$ (x versus t, by matlab)

I am junior in control. If $\dot{x} = Ax$ where $A$ is a $n\times n$ matrix and $x$ & $\dot{x}$ are $n\times 1$ vectors, by $x = \exp(At)$, we can draw $x$ versus $t$. If $\dot{x} = Ax + Bu$, ...
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0answers
18 views

Matlab functions of variables

So I am writing a function to compute the following equations for an SIR model: So here's my code: ...
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2answers
54 views

Question about separation of variables

This is for the heat equation, where $$\frac{\partial U}{\partial t}-k \frac{\partial^2 U}{\partial x^2}=1$$ with the conditions $$U(0,t)=0, \; U(x,0)=0 \text{ and } \frac{\partial U}{\partial t} ...
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1answer
26 views

Hyperbolic Systems ODE

Let $M_n$ the set of matrices of order $n \times n$ identified with $\mathbb{R^{n^2}}$ e $S=\{A \in M_n ; x'=Ax$ is hyperbolic$\}$. Show that $S$ is open and dense $M_n$.
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If at a specific value of x in a non-linear ODE a term is cancelled so that it looks just like the linear ODE, should the result be the same?

If at a specific value of x in a non-linear ODE a term is cancelled so that it looks just like the linear ODE that I am comparing to, should the result be the same (graphic at bottom)? I am ...
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0answers
19 views

Find the streamline

F(x,y,z)=(14x-2y)i +(x^2-y2-z^2)j+(5x*y)k find the stream lines x,y and z i know that i have to find the solution for differential equation: ...
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1answer
16 views

Matlab using subscript variable

I'm trying to write a function in matlab but I don't quite know if it is working. In the equation line i have: xdot(2) = N_h * x; to signify: $$\frac{dy}{dt} = ...
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1answer
64 views

What is the meaning of $\;x\;dx=y\;dy\;$?

I know that $x > y$ and $\;x\;dx=y\;dy.\;$ Can someone explain to me what is the meaning of this and how are x and y related?
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3answers
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Functional equation regarding differentiability

How do you solve this problem ? I'm more interested in the method than the result . Find all the differentiable functions that satisfy the following condition : f(x+y)=f(x)+f(y).
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1answer
21 views

Estimation higher order

Consider non-dimensional differential equation for the height at the highest point is given by \begin{equation} h(\mu)= \frac{1}{\mu}- \frac{1}{\mu^2} \log_e(1+\mu) \end{equation} $0<\mu\ll 1.$ ...
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1answer
19 views

Find the function $y(x)$ with the property that $y(x)$ has a horizontal tangent line

This is a calculus problem I've been struggling on: "Find the function $y(x)$ with the property that $y(x)$ has a horizontal tangent line at the point $(1,-2)$ and $\frac{d^2y}{dx^2}=2x+5$. So, I ...
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1answer
19 views

Steps in the solution of Korteweg-deVries PDE

In the following solution of the Korteweg-deVries PDE $$ u_t + 6uu_x + u_{xxx} = 0 \qquad (3.1) $$ I do not understand the second integration step and how they arrive at the expression for the ...
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1answer
31 views

Solve non-linear ode system as a function of $t$.

I need to solve this ode' system $$ \begin{cases} \dot x=y\\\dot y=-x+x^2=x(x-1) \end{cases} $$ To solve it as a function $x(y)$ or $y(x)$ is trivial, but I need the solution as a function of time: ...
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0answers
18 views

Show that $x' = Ax$ is an attractor if end only if there is a quadratic form $q$ positive definite such that $Dq(x) . Ax < 0$ for all $x \neq 0$

Show that $x' = Ax$ is an attractor if end only if there is a quadratic form $q$ positive definite such that $$Dq(x) . Ax < 0$$ for all $x \neq 0$ Definition: a linear system $x' = Ax$ called ...
4
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2answers
64 views

Show that $e^{t(A+B)} = e^{tA}e^{tB}$ for all $t \in \mathbb{R}$ if, and only if $AB = BA$.

Let A,B real or complex matrixes. Show that $e^{t(A+B)} = e^{tA}e^{tB}$ for all $t \in \mathbb{R}$ if, and only if $AB = BA$. I demonstrated the reciprocal: $\Leftarrow )$ The two equations are ...