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

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Laplace’s equation in the Polar Coordinate System

Laplace’s equation in the Polar Coordinate System: ...
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Logistic equation model.

I need some help on the following question: A population of insects increases at a rate r proportional to the total population. Initially, there are 20000 insects, and birds eat 1000 insects per ...
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1answer
28 views

Solution in common for two differential equations

Consider: $E1: y''-4y'+4y=0$ Solution: $y(x)=c_1 e^{2x}+c_2 x e^{2x} $ $E2: y''-2ay'+(a^2-1)y=0$ Solution: $y(x)=c_1 e^{(a+1)x}+c_2 e^{(a-1)x} $ For what values of $a$, $E1$ and $E2$ have ...
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Jacobi and Gauss Seidel Iteration for solution of ODEs

I have used the Jacobi and Gauss-Seidel iteration schemes for solution of the following ODE: $$y^{''}(x)-5y^{'}(x)+10y(x)=10x $$ I will outline my method below: Discretion the equation by ...
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1answer
33 views

Recommend resources on dynamical systems and singularities

I'm looking for resources on bifurcation theory and systems of non-linear differential equations, but am very particular about the way it is taught/explained. I would like the approach to be based on ...
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0answers
40 views

Counterexample to Peano's theorem in infinite dimension

Would you like a counter example that Peano's theorem does not apply to spaces with infinite dimension. Peano theorem: Let E be a space with finite dimension, consider a point $(t_0,x_0) \in \Re ...
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0answers
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Uniqueness of the solution to a certain IVP

Let $f:[0,1]\to[0,1]$ be a strictly decreasing, continuous function with $f(0)=1$ and $f(1)=0$, and consider the following IVP: $$\frac{dy}{dt}=f(x(t))-y(t), \ \ \ y(0)=0$$ ...
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3answers
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what are the equilibrium points of the following: [closed]

where $x$ represents susceptible individuals, $y$ represents infected individuals. Find the two biologically meaningful equilibria. $$ \frac{\mathrm{d}x}{\mathrm{d}t} =12−3xy−3x $$ $$ ...
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1answer
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Trying to differentte $\ln(|2+f(x)|)=2+e^{x*x}$

I am trying to solve this differential $\ln(|2+f(x)|)=2+e^{x*x}$ so far I did this much; $$ \ln(|2+f(x)|)=2+e^{x*x}\\ |2+f(x)|=e^{2+e^{x*x}}\\ \text{now I have two situations/solutions, because of ...
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1answer
55 views

Bessel Functions Integral Representation proof

So, I'm still working with Bessel functions and trying to proof the following identity, but I'm at a loss for what could possibly be going on here: Any idea how to even approach the proof for ...
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Homogenous Linear ODE with constant coefficients

How do you factor the following Homogenous Linear ODE with constant coefficients and what is the general solution: $$L[f] = \left(\frac{\mathrm{d}}{\mathrm{d}x} ...
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Derivation of Euler Lagrange Equation

I was reading on the derivation of the Euler Lagrange Equations (in the link: http://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation focusing on: "Derivation of one-dimensional Euler–Lagrange ...
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Why do we want that the determinant of the coefficients is $0$?

Eigenvalue problem with periodic boundary conditions-complete Fourier series $$y''+\lambda y=0, 0 \leq x \leq L$$ $$(*): \begin{cases} y(0)=y(L)\\[4pt] y'(0)=y'(L) \end{cases}$$ $$$$ It's a ...
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Equilibrium question [closed]

Consider the differential equation $$x' = x^3 − x^2 − 6x.$$ (a) Find all equilibria. (b) Determine the stability of each equilibrium analytically (not from the phase line diagram). (c) Sketch ...
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1answer
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Solving a PDE: basic first-order hyperbolic equation $u_t+cu_x=0$

So I have to solve the first-order hyperbolic equation $u_t+cu_x=0$ and $c$ as a constant. It is a PDE, since there is the time and spatial variable, but I'm overwhelmed by the maths given in books of ...
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1answer
23 views

Need help with proving a lemma

I need to prove the following with the help of Gronwall's inequality: If, for $t \in [a,b]$, $$\phi(t) \leq \delta_2(t-a) + \delta_1 \int_{a}^{t}\phi(s)ds + \delta_3,$$ where $\phi$ is a nonnegative ...
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20 views

Linear Transformation of Variables

I am wondering if there is some sort of theory/trick that can help me solve this problem: This is for my non-linear dynamics course. We are studying pitchfork bifurcations and the problem is as ...
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1answer
38 views

Need helping proving that something is differentiable but not continuously differentiable

I need some help please proving that a function is differentiable at $(0,0)$ but not continuously differentiable at $(0,0)$. The function is as follows... (from $\mathbb{R}^2$ to $\mathbb{R}$) ...
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1answer
30 views

Ordinary Differentiation $t^2y''=t(t+2)y'-(t+2)y$

$$ t^2y''=t(t+2)y'-(t+2)y $$ The question is how to find the Wronskian without knowing the solutions of this equation? I uploaded the origin question below, which is from a sample test. Anyone ...
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1answer
19 views

When does Initial Value Problems have: no solutions, more than one solution, precisely one solution?

I haven't taking Differential Equations for over 2 or 3 years and it escapes my memory how to determine when would an IVP (Initial Value Problem) would have no solutions more than one solution ...
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1answer
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Laplace's equation-separation of variables

I am looking at the $2$-D Laplace's equation $$\nabla^2u=u_{xx}+u_{yy}=0$$ $$u(x,0)=f(x), x \in (0,a)$$ $$u(x,b)=0, x \in (0, a)$$ $$u(0,y)=u(a,y)=0, y \in (0,b)$$ The solution is in the form ...
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1answer
28 views

Take the Laplace Transform

Take the Laplace transform of $$ \int_{0}^{t}x^2(x-t)^4 \cos(x)dx .$$ I'm not quite sure where to start...
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1answer
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Finding zeroes of a numerical solution of an ODE in Maple

I have a system of ODEs involving many variables, say 20, and I have solved this system numerically by Maple for a particular initial condition. When I plotted these solutions it was clear that ...
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1answer
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Help solving an ODE

This is an example in my book. It is for the following system: \begin{align*} x'&=y+x(1-x^2-y^2)\\ y'&=-x+y(1-x^2-y^2) \end{align*} So using polar coordinates we get the following system ...
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What about uniqueness of general solution?

I found some info about uniqueness for inital value problem. But what about uniqueness of general solution? Is it right that ODE $y'=y$ has two general solutions? 1) $y=Ce^x$ 2) $y=e^{(x+C)}$ Or ...
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1answer
31 views

How to find I(t)?

I'm working with a SIS model for diseases. Where S stands for susceptibles, and I stands for infected. I have a situation that is modeled by the system: $$S'(t)=\frac{dS}{dt}=-\beta SI-\lambda S$$ ...
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1answer
95 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 ...
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21 views

Modification of Gronwall's Lemma

Exercise 2.3 in this book: ...
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0answers
17 views

Proof of Gronwall's Inequality

I have a question about the proof of Gronwall's inequality as given in Chicone: Ordinary Differential Equations with Applications. Gronwall: Suppose that $a<b$ and let $\alpha, \phi,$ and $\psi$ ...
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0answers
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How do impulsive differential equations work? Can you provide an example?

I have heard of impulsive differential equations being used in some epidemiological models of infectious disease. I haven't heard of them before in my math education, and I was wondering how they ...
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Existence of a solution of a nonlinear ODE

I have to show, that the nonlinear ODE $$u'(t)-2u''(t) u(t)=-1,\quad u(0)=1,\,u'(0)=0$$ has a unique solution $v(t)\in C^2(0,T)$ on any Interval $[0,T]$, $T>0$ and that $$\max_{0\leq t\leq ...
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1answer
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Checking the solution to a diffential equation.

Is there a quick way to check that the solution to a diffential equation is correct, I know you can diffentiate it and see if it works but this can take a long time (I want to check my answers in an ...
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1answer
50 views

Finding the Green's function for $y'' + y' = f(x)$

I have this ODE: $$y'' + y' = f(x)$$ with $y(0)=0$ and $y'(1) = 0$. I'm trying to find the Green's function. I multiply through by $G$, integrate over the domain and then use integration by parts to ...
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Are there other well known oscillator systems besides Van der Pol oscillator? [closed]

Is there any collections of oscillator systems similar to "matrix market"?
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Which $n$th order differential equations have $n$ linearly independent solutions?

In these notes (p. 28), it is stated that differential equation $28$ is a second order ordinary differential equation therefore there are two linearly independent solutions. Which is the largest set ...
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2answers
68 views

the global stable and unstable manifolds

Show that $x^* = (1, 2)$ is a fixed point of the system $x_1' = 2 + 3x_1 − 2x_2 − x_1^2 + 2x_1x_2 − x_2^2$ $x_2' = 3 + 4x_1 − 3x_2 − x_1^2 + 2x_1x_2 − x_2^2$ Determine $W^s(x)$ and $W^u(x)$, the ...
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2answers
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Laplace transform using the definition

Find the Laplace of the given function using the definition $$f(t)=tsin(t)$$ I know what the answer is according to a sheet that I have of common transforms but I am not 100% on how to get there ...
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2answers
140 views

Real analysis question involving inhomogenous linear ODE

So I had another problem like this but the ODE was homogenous, now there is a non zero right side. I completed part (i), $\large c(x) = \int \frac{b(x)}{g(x)} dx$. I am stuck on (v). (1) is the ...
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2answers
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Laplace Transform of an integral

Find the Laplace transform of $$f(t)=t\int_0^{t} \tau e^{-\tau}$$ $L(f)(s)$= ?? My thought is that I can change the $\tau$ to $t$ by Transforming the integral to get $$t/s*L[t*e^{-t}]$$ But ...
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1answer
28 views

Laplace transform of integral equation

Use Laplace transforms to solve the integral equation $$y(t)-\frac{1}{2}\int_0^ty(t-v)~dv=1$$ First find the Laplace transform $Y(s)$ of $y(t)$
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1answer
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Convolution of two equations

Find the convolution of $f(t) = t$ and $g(t) = e^{t}$ $$(f*g)(t)= ?$$ If I am correct, I am able to find the Laplace Transform of each individually, then multiply them together. Let $L(x)$ equal ...
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0answers
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interval of solution of a Linear Ordinary Differential Equation with initial conditions

The equation is $$y' + \frac{2ty}{t^2-4} = \frac{2t}{t^2-4}$$ with $y(0) = 1$ as initial condition. What is the solution and its interval? Using some methods of solution I can come up with $y(t^2 - ...
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1answer
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Wronskian Bessel Equations

I need to compute the wronskian of $J_n$ and $Y_n$ (the Bessel functions of the first and second kinds). I've been able to find in many sources that it is $$W(J_n,Y_n)=\frac{\pi}{2x}$$, but I haven't ...
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Using polar form to show that a simple critical point is a spiral point

This is the question in my "homework." I say "homework" because it is not picked up or graded but we are supposed to do it for practice, anyhow here's the question: Given the system ...
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2answers
25 views

Laplace transform of multiplication of three terms

Okay, so I have $${f}(t)= t\mathrm{e}^{-2t}\sin 2t.$$ In order to do a Laplace transform, I'm pretty positive I cannot just split it up cause that would basically break the rules of math. I ...
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0answers
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Show that the D'Alembert operator is a formally self-adjoint operator.

A problem asking me to prove that the D'Alembert operator, defined as: $$\hat\Box^2=\frac{\partial^2}{\partial t^2}-\bigtriangledown^2$$ is a formally self-adjoint operator. To demonstrate the ...
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1answer
33 views

Meaning of $ dx \times dy = k $

Does $ dx \times dy = k $ have a mathematical meaning? What about when considering $y = y(x)$?
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44 views

Question about an eigenvalue problem

I have a question... How can I show that the eigenvalue problem $$y''+λy=0$$ $$y(0)=0,$$ $$ y'(0)=\frac{y'(1)}{2}$$ is NOT a Sturm-Liouville problem?
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1answer
26 views

Which means adjoint problem of a differential equation?

I wanted to know if anyone can help me with the following problem: Get the adjoint problem (differential equation and boundary conditions) for the problem given by: $$\frac{d^2 u}{dx^2}=f(x)$$ ...
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1answer
26 views

Good Source of Differential Equations Problems with Worked Solutions?

I am looking for a good source of problems for differential equations (first order, second order, laplace, convolution, systems). I find it helpful if the question has a worked solution or at the ...