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

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Finding conditions of non existence of Periodic orbit

$$ x'=y \mbox{ and } y'=ax-by-x^2y-x^3 $$ I need non-existence of periodic orbits. Which conditions $a$ and $b$ in $\mathbb{R}$ must satisfy? First, one can see that if $a\leq 0$, then the system has ...
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Differential equations, theoretical question about lowering the power of a basic differential equation.

In the text book it says we can solve:(The area of existence and uniquesness of the equation is $G= R \times R^n$)$$x^{(n)}=f(t), f \in C(R) \tag{1}$$ the following method: Integrating (1) $n$ times ...
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38 views

Eigenvalue problem $y''+py=0$, $y(-2)=0$, $y(2)=0$

The problem states to find the non-negative solutions to the eigenvalue problem given by $y''+py=0$ where p is a parameter which may be varied. Solving this differential equation for the general ...
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22 views

What are the critical values of α where the qualitative nature of the phase portrait for the system changes?

I was given a matrix and solved for the eigenvalues and were marked as correct but I dont know how to solve for the second part of my question. Help please. $r = -1 + \dfrac{\sqrt{100 + 44 ...
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87 views

upper bound of a differential equation solution

Let $A(t)$ be a bounded singular values matrix that is function of time, and $f(t)$ an $L^\infty$ function of time. And consider the ODE $$ \dot x = A(t) x + f(t) $$ How we can describe qualitatively ...
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43 views

Find extremum of functional

I want to find the extremum of $$J(y)= \int_1^2 \frac{\sqrt{1+y'^2}}{x}dx, \ y(1)=0, \ \ y(2)=1$$ I thought to use the following theorem: If $y$ is a local extremum for the functional $J(y)= ...
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22 views

Solving differential equation with Fourier-series-inhomogenity

Let $\lambda$ be a real number , $(c_k)$ a complex sequence with $\mid c_k \mid \leq C(1+\mid k \mid)^{-2}$ for all k with a constant $C \geq 0 $. Find all periodic, two times differentiable ...
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40 views

Continuation of differential equation

Suppose I have a differential equation $$\dot{x} = f(x)$$ which has global solution for any initial value $x(0) \in \mathcal{S}$. Is there some theorem defining conditions under which this equation ...
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35 views

Prove the $f_1, f_2$ is a basis of linear subspace of solution of differential equation

Let $p,q \in C(\mathbb{R}), L_{pq} = \{f \in C^2(\mathbb{R}):f^{(2)} + pf^{(1)} +q f = 0\} $ For each $(a,b)^T \in \mathbb{R}^2$ there is only one $f \in L_{pq}$ with $(f(0),f'(0)) = (a,b)$ 1- ...
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28 views

$V$ is $C^1$ and $V(x_0)=0$ and $ \nabla V $ is not zero $\{ x : V(x)= c \}$ is a surface with no edge around $x_0$

I am studying lyapanov second method in stablity theory of ODE. I have encountered a geometric lemma which says the following: Assume $ V:\mathbb R^n \to \mathbb R$ is a $C^1$ and $x_0 \in \mathbb ...
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29 views

Division of two series expansions

I have the two functions $u(x)$ and $v(x)$, both of which have known basis expansions $u(x) = \sum_n a_n f_n(x)$, $v(x) = \sum_n b_n f_n(x)$. I would like to calculate the function ...
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81 views

Certain Lie algebra structure on $\chi^{\infty}(\mathbb{R}^{2})$ or $\chi^{\infty}(S^{2})$

Is there a lie algebra structure $ [ \;. ] $ on $\chi^{\infty}(\mathbb{R}^{2})$ or $\chi^{\infty}(\mathbb{S}^{2})$ which is not isomorphic to the standard structures but satisfies the following: ...
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33 views

Connection between possibility of non-monotonic solutions to first-order delay differential equations and 1-d discrete dynamical systems?

Is there a connection between the possibility of non-monotonic solutions, including periodic or other oscillatory solutions, arising in first-order autonomous delay differential equations such as the ...
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12 views

Is there a test for tractability of nonlinear differential equations?

After lengthy attempts at tackling the problem one might say that coming up with a closed form solution for a nonlinear differential equation is not possible - that the problem is intractable. But is ...
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34 views

ordinary differential equations

I am trying to understand how the solution of this equation goes: $$\frac{y^2-1}{y}\cdot \sin(x^3)=\frac{dy}{dx}$$ with initial condition $y(0)=-0.5$ I would like to understand if the solution can ...
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23 views

Clarification of Fuchs's theorem

Here is Fuchs's theorem My professor has been saying the last couple of classes that if $p(t)$ and $q(t)$ are polynomials, then the second order differential equation converges everywhere. He hasn't ...
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154 views

What are all types of elementary second order ordinary differential equation that can not be expressed in closed form?

Can we define all types of elementary second order ordinary differential equation that can not be expressed in closed form as opposed to the one that we can solve? In differential algebra, ...
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41 views

General solution of $ty'+2y=4t^2$

Should we left the general solution of the differential equation $t\frac{dy}{dx}+2y=4t^2$ as $t^2y=t^4+c$ instead of $y=t^2+c/(t^2)$ ($c$ is an arbitrary constant)? Does the solution $y=t^2+c/(t^2)$ ...
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53 views

PDE question: heat equation (third order??)

I am familiar with the usual heat equation, however, my lecturer gave me this problem and it does not look like anything I have ever seen (in my whole entire life and I am not just being dramatic). ...
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Equillibria to Differential Equations

I am wondering what the exact definition is of an equilibrium to a differential equation. It seems like the general consensus implies that a differential equation will only have an equilibrium if it ...
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37 views

Asymptotic Behavior of Differential Equation

physicist here. I'm studying some problems that involve the use of differential equations. The professor of the course has indicated that usually variable changes used to simplify the equations come ...
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17 views

For what types of differential equations is the Laplace transform most effective?

I'm reviewing for a final exam and want to make sure I know what tools to use for what situations, and was just wondering if there were situations where the Laplace transform is unusable or less ...
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26 views

Partial Differential Equations Black Scholes Problem

Part 1) Consider the Black-Scholes problem $$\frac{\partial A}{\partial t}+\frac{\sigma^2B^2}{2}\frac{\partial^2A}{\partial B^2}+rB\frac{\partial A}{\partial B}-rA=0 ...
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ODE from systems biology, can I generalize this? Have solution but not sure how to arrive at it.

Reading a systems biology book, and it describes a model with the following ODE: $$ \frac{dY}{dt} = -\gamma Y + v_1 X_1 (T - Y) + v_2 X_2 (T - Y)$$ where $Y$, $T$, $T - Y$, $X_i$, $a$ and $v_i$ are ...
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57 views

Is there an elegant proof of this elementary bifurcation theory result?

Let's suppose I have a $C^1$ function $f:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$, $(x,\lambda)\mapsto f(x,\lambda)$. Suppose there is a unique solution of the equation $f(x,\lambda_1)=0$, ...
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27 views

Find the points at which an IVP admits at least two solutions

Given the IVP: $$\frac{dy}{dx} = x + |\sin(y)|$$ $$y(x_0) = y_0$$ Find the points in $\mathbb R^2$ at which this IVP admits at least two solutions. Clearly, $f(x,y) = x + |\sin(y)|$ is Lipschitz ...
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93 views

Solving second order nonhomogeneous differential equation with non-constant coefficients using Laplace Transform

$ty''(t) + y'(t) -ty(t)= tf(t)$ How to solve the problem using Laplace Transform? Using Laplace transform I got $$Y(s)= C(s^2-a^2)^{-1/2} + (s^2-a^2)^{-1/2}\int (s^2-a^2)^{-1/2}F(s)\,ds$$ where ...
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How can one prove the existence and uniqueness of solutions to linear differential equations?

It is a theorem (I think) that the equation: $$\mathbf{x}'(t) = A(t)\mathbf{x}(t) + \mathbf{b}(t); \qquad \qquad \mathbf{x}(t_0) = \mathbf{x}_0$$ Has a unique global solution for any matrix ...
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158 views

System of first order ODEs with coherent sinusoidal time varying coefficient

I have encountered equations of the form $$\frac{{d{\bf{y}}(t)}}{{dt}} = \left( {{A_0} + {A_1}\cos (\omega t)} \right){\bf{y}}(t)$$where ${\bf{y}}$ is a vector and ${{A_0}}$ and ${{A_1}}$ are square ...
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50 views

Differential Equation $(1+x^2)y'-2xy=\cot(x)$ after integrating factor

$$(1+x^2)y'-2xy=\cot(x)$$ or $$y'=\frac{2x}{1+x^2}y+\frac{\cot(x)}{1+x^2}$$ if I use an integrating factor $(e^{\int\frac{-2x}{1+x^2}dx}=\frac{1}{1+x^2})$ I get $$\frac{y}{(1+x^2)}=\int\frac{\cot ...
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Separable equation

I am looking at this first order separable differential equation, and I am stuck. Here is the equation: $\frac{du}{dt}=u$ I seperated like this: $ \frac{du}{u}=dt $ Integrated both sides and ...
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37 views

Solving differential equation with small parameter

I am trying to solve the following equation $$\frac{x}{x+1}\frac{d^{2}\left(\phi^2\right)}{dx^{2}}+\frac{2x+1}{(x+1)^{2}}\frac{d\left(\phi^2\right)}{dx}=\frac{1}{3\phi}$$ with $x\ll1$ and ...
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Is one dimesional cubic NLS is globally wellposed in $H^{s}(\mathbb R), (0<s<1)$?

We consider the one dimensional cubic nonlinear Shr\"odinger equation (NLS): $$i\partial_{t}\phi (x,t) +\Delta \phi (x,t)= \pm |\phi (x,t)|^{2} \phi(x,t), \ (x, t\in \mathbb R),$$ $$\phi (x,0) = ...
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30 views

Zeroes of a solution to a differential equation

Show that any solution to the equation $y''+xy=0$ has at least 15 zeroes on the interval $[-25,25]$. Please give me a hint.
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Is the following complex value differential equation always has a solution?

Let $a(z)$ be a fixed complex value complex variable function, not necessarily holomorphic. Consider the following differential equation $$ \frac{\overline{\partial}f}{\partial \overline{z}}+af=0. $$ ...
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Linear ordinary differential equations and their evolution operators for measurable operators

Consider the following homogeneous IVP: $$\begin{cases} \dot{u}(t)+A(t)u(t)=0 \\ u(0)=u_0 \end{cases}$$ for $u:[0,1]\to \mathbb{R}^n$ (some interval to some finite dimensional Hilbert space, let's ...
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37 views

Differential equation of the form to find

Lets $f(z)$ is some analytic function on complex plane and $y(z)$ is known analytic function on complex plane. Problem statement: find all $f(z)$ that: $$f(z) = f(z\frac{\partial}{\partial z})y(z)$$ ...
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33 views

Proving the Bessel function solves the Bessel equation

Using the notation for the Bessel function as $J_n(z)=\sum \limits_{k=0}^{\infty}\frac{(-1)^kz^{n+2k}}{k!(n+k)!2^{n+2k}}$, I want to show that $w=J_n(z)$ satisfies ...
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117 views

Periodic Solution of Damped Pendulum with Constant Torque

I have a system of ordinary differential equations $ \theta' = v$ $ v' = -bv - \sin \theta + k$ These are the equations for a pendulum with $\theta$ being angular position, and $v$ being angular ...
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86 views

Eigenvalues for $y''+2y'=\lambda y$

I must find the eigenvalues and eigenfunction for $$y''+2y'=\lambda y$$ with initial conditions $y(0)=0$, $y'(1)=0$. I have found the non-trivial case, and made an attempt to solve for $\lambda$, but ...
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Eigenvalue Function of Laplace Equation discretizes by nine-point stencil

I'm trying to plot the eigenvalue function of the Laplace equation $$-u_{xx}-u_{yy}=0,\;(x,y)\in (0,1)^2$$ with $$u(x,y)=0$$ on the boundary of the unit square. I have the nine-point stencil ...
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61 views

Using Green's Function to Find Particular Solution

We have the non-homogeneous differential equation $x^3y'''-3x^2y''+6xy'-6y=4x^2$ with conditions $y(1)=1, y'(1)=1, y''(1)=0$, and I have been tasked with finding its particular solution using Green's ...
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73 views

Show the IVP has a unique solution

Assume that $f:\mathbb{R}^{n}\times \mathbb{R} \to \mathbb{R}$ satisfies (i) there exists a constant $M>0$ such that $|f(x,t)-f(y,t)|\leq M|x-y|$ for each $x,y\in \mathbb{R}^n$ and each $t\in ...
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For time varying linear ODEs: is there a transformation which can make the system admit an exponential solution?

First, I'll start with the properties of the matrix in question: Assume we are given some matrix $\mathbf{A}(t)$, which is time dependent. This matrix is square and not invertible. A system: $$ ...
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48 views

Constructing a function using the Fourier transform

Pick an integer $n\ge 5$ and let $f\in C_{C}^{\infty}(\mathbb{R}^{N})$. We want to use the Fourier transform to formally construct a function $u\in L^{\infty}(R^{n})$ that solves ...
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Leading behaviour of DE at infinity

This is taken from the book of Bender and Orszag, problem 3.44. Find the leading behavior as $x\rightarrow+\infty$ of the differential equation: $x^3y'' - (2x^3 -x^2)y' +(x^3-x^2-1)y=0$ Explain ...
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What problems are related with the following type of FDE with delay?

Consider the following class of functional differential equations with delay: $$\begin{align} \frac{du}{dt} &= F(x,t,u(x,t),u_{t,x}), & (x,t) &\in [a,b] \times [0,T] \\ u(x,t) &= ...
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38 views

Why does an infinite Neumann boundary condition become a Dirichlet condition?

Often when I read a paper I see a statement of the type: Our boundary condition at the surface is $\frac{\partial f}{\partial x} = \alpha$. In the limit of $\alpha \to \infty$ this is equivalent ...
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31 views

Writing a 2nd order linear ODE as a set of 2 first order ODEs in its independent solutions

Given the homogeneous linear ODE $$y''(x)+ P(x) y'(x) + Q(x) y(x)=0$$ where $x\in(0,\infty)$ and $P$ and $Q$ are some smooth (but not necessarily bounded) functions. I know that we can write this as a ...
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66 views

How many solutions does Riemann's P-symbol describe?

The Papperitz-Riemann P-symbol $$ \tag 1 y(z) = P \left\{ \begin{matrix} z_1 & z_2 & z_3 & \; \\ \alpha_1 & \alpha_2 & \alpha_3 & z \\ \beta_1 & \beta_2 & ...