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

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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: Right now, I'm speculating that it has something to do with ...
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72 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 (iii) and the ...
4
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1answer
37 views

How to show no periodic orbits exist

I am trying to show that no periodic orbits exist for the system: $$ x_1'=y+x^2+xy^3$$ $$y'=-2x-y^3$$ I have tried using Dulac's criterion to find a function $g(x,y)$ such that $\Phi(x,y)$ given by ...
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1answer
13 views

Laplace’s equation in the Polar Coordinate System

Laplace’s equation in the Polar Coordinate System: ...
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1answer
9 views

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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16 views

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
22 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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3answers
33 views

what are the equilibrium points of the following: [on hold]

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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9 views

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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9 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
33 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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2answers
36 views

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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0answers
16 views

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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0answers
8 views

Differential equation with variable change of function composition

I need to find all the functions $z(x)$ for $z'-e^{x^3} cos(x) z = 3x^2 z L(z)$ As sugerence, i have a proposed variable change: $y(x)=L(z(x))$ To do that var change, i need to express the equation ...
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1answer
33 views

How can I solve an ODE when $F(x_0)=F'(x_0)=0$ is given at an unknown point $x=x_0$ using bvp5c?

I'm attempting to solve the following ODE using MATLAB bvp5c. I've used bvp5c for other typical multipoint boundary value problems but I have no idea how to deal with ODEs with conditions given at an ...
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1answer
10 views

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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2answers
63 views

linear differential equation problem [closed]

Consider the following system of linear differential equations: $$\begin{split} \frac{dx}{dt}&=−3x+y\\ \frac{dy}{dt}&=x−3y \end{split}$$ Find the eigenvalues and eigenvectors associated ...
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1answer
34 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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0answers
15 views

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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1answer
47 views

Differential equation question

Consider the differential equation $\dfrac{dx}{dt}=x^3−x^2−6x$ . Find all equilibria. Determine the stability of each equilibrium analytically (not from the phase line diagram). Sketch the ...
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1answer
22 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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1answer
12 views

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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9 views

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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Equilibrium question [on hold]

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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18 views

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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25 views
+50

second order linear ode in the complex domain

Consider $w''(z)+p(z)w'(z)+q(z)=0$ where $p(z), q(z)$ are analytic for $R\le|z|<\infty$ for some fixed $R$. Now I want to prove using analytic continuation of the solutions that the ode has one ...
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19 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 ...
2
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2answers
56 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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1answer
36 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
96 views

Find values of the parameters in Predator prey model

$$r' = F_1(r, f) = r − cr^2 − drf$$ $$f' = F_2(r, f) = −f/4 + erf + gf^2$$ Consider the case where $g = 0$. For what values of the parameters, $c, d$ and $e$, which are all assumed to be positive, ...
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1answer
17 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 ...
2
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1answer
23 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
26 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
30 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: $$S'(t)=\frac{dS}{dt}=-\beta SI-\lambda S$$ ...
3
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1answer
22 views

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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0answers
29 views

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
79 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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19 views

Modification of Gronwall's Lemma

Exercise 2.3 in this book: ...
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1answer
50 views

What is the best approach to solve $ 4y^3 y''=16 y^4 -1$?

How can I solve this DE: $$ 4y^3 y''=16 y^4 -1$$ I really would not bother asking if Wolfram alpha had not exceeded comp. time and not shown me step-by-step solution.
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1answer
291 views

Show that a set is positively invariant set [on hold]

Consider the system: $$\frac{dx_1}{dt} = -x_1$$ $$\frac{dx_2}{dt} = (x_1x_2 - 1)(x_2)^3 + (x_1x_2 - 1 + (x_1)^2)x_2$$ Show that $T= \{x\in \Bbb{R}^2\mathbin{|}x_1x_2 \geq 2\}$ is a positively ...
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0answers
16 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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1answer
38 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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0answers
64 views

Pencil of conics and periodic orbits

Let $\dot{x}=P(x,y)$ and $\dot{y}=Q(x,y)$ be a quadratic polynomial differential equation. Prove that if the pencil of conics $P+\lambda Q$ contains an imaginary conic, a real conic reduced to a ...
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0answers
9 views

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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1answer
16 views

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 ...
2
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0answers
23 views

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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2answers
21 views

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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7 views

Obtaining characteristic v on Cauchy Problem

$(x-y)p+(y-x-z)q=z$ Find the integral surface which the curves it passes are $z=1$ and $x^2+y^2=1$ Here is my try. $$\frac{dx}{x-y}=\frac{dy}{y-x-z}=\frac{dz}{z}$$ So we have ...
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1answer
26 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
24 views

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 ...