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

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State transformation for non-holonomic differential equation

Given a non-holonomic dynamical system, \begin{align*} \dot x = vcos\theta \\ \dot y = vsin\theta \\ \dot \theta = \omega \end{align*} with constraints $|v| < v_{max}, |\omega| < \omega_{max}$, ...
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Finding when fixed point is hyperbolic

Consider the IVP for the $2$-dimensional dynamical system ($X=[0, \infty )^2$) $$\dot{x_1}=a-x_1-\frac{4x_1x_2}{1+x_1^2}$$ $$\dot{x_2}=bx_1 \bigg( 1- \frac{x_2}{1+x_1^2} \bigg)$$ for all $t \in I$, ...
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0answers
7 views

Critical point of an ODE

I have been asked to deduce if an ODE has a critical point from drawing its isoclines and then sketching the integral curve. What exactly is a critical point of an ODE, and how would I deduce it from ...
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0answers
10 views

Help following a published calculation of a prediction confidence interval for a prediction made from a simple ODE

I'm trying to follow a calculation made in a paper(section 2 from the supplementary contents of ...
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0answers
10 views

Finding a Lyapunov function for system of non-linear differential equations

I have the following system of differential equations and I would like to find a Lyapunov function for it, in order to classify the point $(0,0,0)$: $$ \begin{cases} x' &= -y-xy^2+z^2-x^3,\\ y' ...
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1answer
25 views

Differential equation: $e^{xy^2}\frac{x}{x^2+1}\,dx - e^{y^2x}\,dy = 0$

I have no idea how to solve this and would love some help. $$e^{xy^2}\frac{x}{x^2+1}\,dx - e^{y^2x}\,dy = 0$$
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2answers
29 views

Vector norm and relationship with euclidean distance

If $y\in E_n$ (n dimensional euclidean space) show that $||\textbf{y}||\leq|\textbf{y}|\leq \sqrt{n}||\textbf{y}||$ Where $||\textbf{y}||$ is the euclidean length of the vector $\textbf{y}$ and ...
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3answers
27 views

What exactly is the maximal solution of an ODE and why do we care?

I am reading these notes on the definition of a maximal solution of an ODE i.e. http://www.math.lmu.de/~philip/publications/lectureNotes/ODE.pdf But the definition is sooo abstract and no example is ...
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0answers
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Finding a lower and upper bound for the first eigenvalue of a Sturm Liouville problem

Given the eigenvalue problem $y''+\lambda (1+x^2)y=0$, $y(0)=y(1)=0$, I need to find a lower and upper bound for the first eigenvalue $\lambda_0$ (that fits the eigenfunction that has no zeros in ...
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1answer
7 views

Understanding the steps taken in a calculation of the maximum profile likelihood of a simple ODE, given some data

I'm trying to understand a calculation made in a paper (section 2 from the supplementary contents of ...
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24 views

Is there a close-form solution for the non-linear difference equation? [on hold]

is there a close-form solution for the difference equation below? $$(x_{n+2}-x_{n+1})-(x_{n+1}-x_n)=(\frac{x_{n+1}}{c})(x_{n+2}-x_n)$$ Any comments are appreciated.
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Dermining stable and unstable manifolds - is my result ok?

Determine all stable and unstable manifolds of the equilibria of $$ \dot{x}=x(1+x)(1-x). $$ Are there homoclinic/ heteroclinic solutions? Hey, just would like to know if I am ...
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0answers
10 views

Quasi-linear partial differential equations. Solving them.

This is what I have as a quasi-linear partial differential equation:$$u(x_1,...,x_n), \ \ \ \ \sum_{i=1}^{n}A_i(X,u) \frac{\partial u}{\partial x_i}=A_{n+1}(X,u) \ \ \ (1)$$ Then it says let ...
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2answers
34 views

How can I solve a first order ODE with $\pm$ signs by the Integrating Factor method?

I have the following first order ODE to be solved via the integrating factor method: $$\frac{\mathrm{d}z}{\mathrm{d}y}\pm z=-\frac12y\tag{1}$$ This is in the general form: ...
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0answers
16 views

Methods for first order PDEs in higher dimensions

What are the possible known methods for solving first order PDEs in higher dimensions? Is there anything else besides the method of characteristic curves? In particular, I have four first order, ...
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1answer
13 views

Show exponential stability quadratic form

Please help me with the following proof: Suppose $\dot x=f(x(t))$ and suppose that we have: $$ \frac{d}{dt}\left( x(t)^TPx(t) \right)\le -x(t)^TQx(t) $$ where $P$ and $Q$ are symmetric ...
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0answers
25 views

If a and b are negative , then can we use the same method we are taught for solving the equation y'' + ay' + by=0 ,

If $a$ and $b$ are negative , then can we use the same method that we are taught for solving the ODE which is $$y'' + ay' + by=0$$
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1answer
17 views

Solving Laguerre coefficients with Integral?

I'm having some difficulty understanding the solution to a particular Laguerre expansion. The problem reads "Expand the term $ e^{-x}$ as a Laguerre expansion, noting the orthogonality of $$ < ...
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0answers
29 views

Transient Terms in a General Solution

Find the general solution of the given differential equation: $$ (x^2-4)(\frac{dy}{dx}) +4y = (x+2)^2 $$ I found the general solution of the D.E and I got the following correct solution: $$ y = ...
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1answer
25 views

PDE Proof that a linear combination of 2 solutions is also a solution [on hold]

Can someone please help? I've been trying to figure this for a few days now. Consider the first order PDE: $au_t + bu_x$ = 0, where a and b are constants. Show that if $u_1$ and $u_2$ are solutions ...
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How to integrate to solve a PDE with mixed partials in the integrand

Problem Statement: Determine the equlibrium temperature distribution inside a circular annulus $r_1\leq r \leq r_2$. If the outer radius is at temperature $T_2$ and inner radius at temp $T_1$. So ...
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0answers
17 views

What is the largest t-interval on which guarantees a unique solution? [on hold]

What is the largest t-interval on which guarantees a unique solution for this equation? $$y'' + y'+ 3ty = \tan t,\quad y(\pi) = 1,\quad y'(\pi) = -1$$
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Solving $\vec y'=\begin{pmatrix} 1 & -1 \\ 0 & 1\\ \end{pmatrix}\vec y$

I need to solve $\vec y'=\begin{pmatrix} 1 & -1 \\ 0 & 1\\ \end{pmatrix}\vec y$. The characteristic polynomial is $(r-1)^2$, so the only eingenvalue is $1$. I found ...
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1answer
27 views

Showing that if $y(x)$ is a solution, then $y(-x)$ is also a solution for a specific ODE

Given the ODE $(1-x^2)y''-xy'+\alpha^2 y=0$, I need to show that if $y(x)$ is a solution, then $y(-x)$ is also a solution. From what I understand, because $y(0)=y(-0)$, it means that all solutions are ...
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0answers
33 views

Different answer when using the 'method of undetermined coefficients' compared to Laplace transform

I have an ordinary differential equation: $$ \frac{\mathrm{d}^2u}{\mathrm{d}t^2} + u = \mathrm{e}^{-t}\cos(t)$$ with $u(0) = u_0$ and $\dot{u}(0) = v_0$, when using the method of undetermined ...
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1answer
20 views

How to calculate the variation of a matrix?

Suppose we have two diagonal matrices $$ A_{\mu \nu}=\left(\begin{array}{cccc} \rho(t) & 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0 ...
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1answer
17 views

Wronskian of two independent solutions equaling zero at a specific point only?

Given $y_1(x)=\sin(x^2)$ and $y_2(x)=\cos(x^2)$, I constructed a linear, homogenic ODE of order 2 by solving: $$ \begin{vmatrix} y & y_1 & y_2 \\ y' & y_1' & ...
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2answers
29 views

Find all the solutions of the initial value problem for a first order non-linear equation

I am trying to solve the initial value problem: $$ y'= \frac{10}{3}xy^{2/5}, \qquad y(0)=0 \qquad \qquad (1) $$ where $ x\in \mathbb{R} $. The first order equation is not linear in the form: $$ ...
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1answer
29 views

What is the dimension of set of all solutions to $y''+ay'+by=0$?

$$y''+ay'+by=0,\quad y(0)=y(1)$$ where $a$ and $b$ are positive real numbers. Let $V$ be the set of all the solutions of this equation. Then the dimension of $V$? The answer to it that I think is ...
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1answer
23 views

Non-monotonically decreasing flow whose limit is $\vec{0}$

I'm trying to come up with $x'=Ax$, which is a system of linear differential equations, whose flow satisfies $\lim\limits_{t\to\infty} \lvert e^{tA}x\lvert = 0$ for all $x\in \mathbb{R}^n$, but ...
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1answer
45 views

Compare analytic model with numerical, mass spring system.

So I'm trying to solve a problem here and I have been working on it all day, clearly i'm in need of some guidance. I have a rod of length $L$ and cross section area $A$, Young's modulus $E$ and ...
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1answer
32 views

Why do difference of squares partial fractions have to be decomposed this way?

Why do you have to factor out $-1$ here? $$\frac{2000}{(10-h)(10+h)}$$$$=\frac{A}{10-h}+\frac{B}{10+h}$$ Decomposing this finds A annd B to be 100, which is wrong. Symbolab and Wolfram Alpha factor ...
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0answers
22 views

method of characteristics for non-linear PDE

I'm trying to solve the PDE $u_x^2-u_y^2=8u$ with initial conditions $u(x,x)=f(x)$. I have that $F(x,y,u,p,q)=p^2-q^2-8u$, with $p=u_x, q=u_y$, and then \begin{equation*} \begin{array}{ll} ...
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Peano theorem expression. [on hold]

How is the Peano theorem expressed in analogous terms to the Lipschitz condition: $|f(x,y_2)-f(x,y_1)| > M|y_2-y_1|?$
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Let $y$ be the solution of $y^\prime +y= \mid x\mid,~~x \in \mathbb{R},~~ y(-1)=0$ then $y(1)=$ [on hold]

Let $y$ be the solution of $y^\prime +y= \mid x\mid,~~x \in \mathbb{R},~~ y(-1)=0$. Then $y(1)=$ (a) $ \frac{2}{e}- \frac{2}{e^2}$ (b) $ \frac{2}{e}- 2e^2$ (c) $2- \frac{2}{e}$ (d) $2-2e$
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2answers
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Solution of differential equation with complex coefficients

Given $$\dfrac{d^2y}{dx^2}-(3-2i)y=0,\quad y(0)=1,\quad y(x\rightarrow \infty)\rightarrow 0$$ then what is $y(\pi)$ ? The answer given is $-e^{-\pi}$. But I cannot understand how its solution can ...
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0answers
37 views

Solve this system of nonlinear differential equations

Although I have a brief understanding of solving linear ODE systems, I got stuck with this non-linear system: \begin{align*} \left(x'(t) \right)^2 + k_1 x + k_1 y = 0\\ \left(y'(t) \right)^2 + k_2 x ...
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0answers
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solve $4x^2y'' + y=0$, $y(-1)=2, y'(-1)=4$

This would require taking the $\ln(-1)$, which Zill solved in the 7th edition of diff eq $4.7$ problem $37$ by substituting $t$ for $x, y(1)=2, y'(1)=4$. Then substituting $-x$ for $t$ in the final ...
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1answer
32 views

Solve the differential equation : $0.5 \frac{dy}{dx}=4.9-0.1y^2$

The question is to solve the differential equation : $$0.5 \frac{dy}{dx}=4.9-0.1y^2$$ What I have attempted: $$0.5 \frac{dy}{dx}=4.9-0.1y^2$$ $$ \frac{dy}{dx} = \frac{4.9-0.1y^2}{0.5} ...
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1answer
28 views

System of differential equation (Matrix form)

I'm trying to solve this system $$ M\ddot{X}(t) = KX(t) $$ where M is a known diagonal matrix and K is a symmetrical known matrix. I'm asked to do the ansatz $Y(t) = M^{1/2}X(t)$ where $M^{1/2} = ...
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1answer
26 views

A specific question regarding a proof in Hassan Khalil's book, Nonlinear Systems

I am trying to understand the proof of a Lemma in the book 'Nonlinear Systems' by Hasaan Khalil (3rd edition). In the Proof of Lemma 3.1, about Lipschitz continuity of vector valued functions, I am ...
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2answers
25 views

Linear-Homogeneous vs Homogeneous ODEs?

Currently in my third week of my first ODEs class and I've already encountered something I'm struggling with. My second homework assignment requires me to classify and solve some ODEs. He gave us four ...
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1answer
29 views

Bifurcation Diagram question for Population harvesting model $P' = rP (1-\frac{P}{K}) - hP$

A deer population grows logistically and is harvested at a rate proportional to its population size. The dynamics of population growth is modeled by $P' = rP (1-\frac{P}{K}) - hP$ where $r$ (the ...
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2answers
44 views

Effect of wronskian on the solution of a differential equation

As far as my understanding goes, the Wronskian $W(t)$ for a second order homogenous differential equation with continuous coefficients can help us govern whether the solutions will be linearly ...
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3answers
38 views

How might I go about forming a general solution for the following differential equation? [on hold]

$\frac{df}{dt}+t^kf=t^k$ where $k\in \mathbb{Z}$ I've solved for the equation previously where $k=2$ to get $f=\frac{-1}{t-t\ln t}+c$ but am not sure how I should go about solving this generally.
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1answer
24 views

Maximal Solution to Differential Equation

For the differential equation $$\dot x = x(1-x), x(0)= \frac 12$$ Decide if the solution exists for all $t \ge 0$ or only on a finite time interval $0 \le t \lt T$. By the theorem, for the maximal ...
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2answers
38 views

Differential Equations $ v \frac{dv}{dx} = -g \frac{a^2}{x^2}$

Question: A particle is projected vertically upwards from the Earth's surface. Its distance $x$ from the centre of the Earth is connected with its upwards speed $v$ by the differential ...
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0answers
37 views

Differentiating a matrix product

In one of the books I found that given that for a linear system $x'=Ax$, there exists a matrix $Q:=\int\limits_0^\infty B(t)dt$, where $B(t)=e^{tA^T}e^{tA}$, and $V(x) = x^T Q x$, ...
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1answer
43 views

Singular solutions of a system of nonlinear 2nd order ODEs

I'm faced with the following nonlinear 2nd order system of ODEs: $$ \phi''(r)+\frac{4r^3-1}{r^4-r}\phi'(r)+\frac{r^2 h(r)^2+2r(r^3-1)}{(r^3-1)^2}\phi(r)=0, \\ ...
2
votes
0answers
45 views

A kind of Sturm-Picone theorem?

My question is very simple: Suppose $u,v:(a,b)\subset \mathbb{R} \to \mathbb{R}^+$ solve \begin{equation} (p(x)u'(x))'=-q(x)f(u(x)) \end{equation} \begin{equation} (p(x)v'(x))'=-r(x)g(v(x)) ...