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

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What is the solution to the system $\frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1}$?

I'm trying to solve the system $$ \begin{matrix} & \frac{df_1}{dt} = kf_1+lf_2 \\ & \vdots \\ & \frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1} \\ & \vdots \\ & \frac{df_N}{dt} = ...
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26 views

Derivative of terminal state w.r.t. the inital conditions.

Let $x\in R^n$ and consider the system $$ \dot{x}=f(t,x) \;\;\mbox{with}\;\; x(0)=x_0 $$ and suppose that we know it's exact or very accurate solution $x(t)$ for the time interval $[0,T]$. I'm ...
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64 views

Please identify this equation: $\nabla^2 \mathbf F -k^2 \mathbf F = \mathbf A$

Is this equation $$ \nabla^2 \mathbf F -k^2 \mathbf F = \mathbf A $$ somehow named? F and A are vector fields. I guess inhomogeneous sign reversed Helmholtz equation isn't appropriate ...
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37 views

Stability properties of discretization of ODE

I am trying to find some conditions which guarantee that a continuous time dynamical system and it's discretization have the same behavior with regard to equillibrium points. Specifically that if the ...
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37 views

How to solve the following an ODE?

Let $x,y,z$ be a given point in $\mathbb{R}^3$. How to solve $(x'(t),y'(t),z'(t))=(x(x+y+z), y(x+y+z),z(x+y+z))$?
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128 views

Eigenvalues problem for generalized Kuramoto-Sivashinsky equation

I been working on Kuramoto-Sivashinsky Equation. In the process of analysis, I need to solve the following eigenvalues problem $$ -u_{xxxx}-\lambda u_{xx}=\beta(\lambda)u $$ where $\lambda$ is a ...
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50 views

ODE particular solution (physics)

I have to do this exercise: ($Z(t)=I(t)$, it's printed wrong). I have a doubt about the first item. To find all resonance when $R=1$, I found the particular solution $I_{p}(t)=A\sin(\omega ...
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Computation of the Frenet-Serret trihedron in $\Bbb L^3$ (Lorentz-Minkowski space)

Consider $\Bbb L^3 = (\Bbb R^3, \langle , \rangle)$, with the convention $$\langle (x_1,y_1,z_1), (x_2,y_2,z_2)\rangle = x_1x_2+y_1y_2 - z_1z_2$$ and $\| v \| = \sqrt{|\langle v, v \rangle|}$. Let ...
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For all topological conjugation $$h: \Delta_1 \rightarrow \Delta_2$$ we have to $h(\omega(p))=\omega(h(p))$, for all $p \in \Delta_1$

Let $X_1$ and $X_2$ fields in $\Delta_1,\Delta_2$ subset open in $\mathbb{R}^n$. Then, for all topological conjugation $$h: \Delta_1 \rightarrow \Delta_2$$ we have to $h(\omega(p))=\omega(h(p))$, for ...
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60 views

How to solve this complicated differential equation?

I need to know how to solve this complicated differential equation in $z$ either analytically or numerically : \begin{eqnarray} \frac{dx_1}{dz} &=& -ib_1x_1 - ikx_2 \\ \frac{dx_2}{dz} ...
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52 views

General solution of ODE

please what is the general solution of $$-(p(t)u')'+q(t)u=0$$ where $\displaystyle\frac{1}{p},\frac{1}{q}\in L^1((0,+\infty))$ Thank you
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Solution to Schrödinger equation $ \partial_t f(x,t) = -\partial_x^2 f(x,t) + \delta(t)V(x)f(x,t).$

I want to solve $$ i\partial_t f(x,t) = -\partial_x^2 f(x,t) + \delta(t-t_0)V(x)f(x,t),$$ for any $V \in C^{\infty}[-1,1]$ and $f: [-1,1] \times \mathbb{R_{\ge 0}} \rightarrow \mathbb{C}$. I would ...
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43 views

Power series to solve differential equations?

We can use the formula $$F(x)=e^{λx} [ ρ-λμ-\dfrac{1}{2} λ^2 σ^2 ]^{-1}. (1) $$ to derive an expression for F(x) when f(x) is any integer power $x^n$. Begin by observing that for the ...
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154 views

Is two-body motion planar?

The two-body problem studies the motion of two bodies under the influence of their gravitational attraction. Following the notation used in Wikipedia http://en.wikipedia.org/wiki/Two-body_problem, ...
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42 views

Differential Equations and Eigenvalues

I have the following system of differential equations: $$\left\{\begin{aligned} \frac {dx} {dt}=-4x+2y \\ \frac {dy} {dt}=-\frac 5 2x+2y \end{aligned} \right. $$ Which corresponds to the following ...
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155 views

Solving an infinite non autonomous system of differential equations.

For all $\lambda\in\mathbb{R}$, let $J(\lambda)$ be the infinite matrix where $(J(\lambda))_{nn}=\lambda$, $(J(\lambda))_{n,n+1}=1$ for all $n\in\mathbb{N}$, and all other entries are $0$. This matrix ...
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29 views

Derivation of the prolongation formula for finding symmetries of diff equations from Olver

I am having a problem with the derivation of the prolongation formula from PJ Olver's text :"Applications of Lie groups to differential equations" Page 105,106. Considering a differential equation ...
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Stability analysis of a three-dimensional system

Study the stability of the equilibrium point $(y,q,z)=(0,0,0)$. (Hypothesis: $\nu,\theta,\zeta$ are positive.) $$\begin{align} \dot{y}&=y(1-\nu -\theta -y-z+\theta y-(\nu +\zeta)q)\\ ...
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How to solve this system of inhomogeneous differential equations

In some past exam papers for the Maths course that I attend,I found this example and I would really appreciate if someone looked at my solution. It goes like this: Find general solution to $$ y_1' = ...
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Tips to find the second solution for homogenous second order ODE

Suppose we have some second order ODE $p(x)y''(x) + q(x)y'(x) + r(x)y(x) = 0$ and that the coefficients are such that we can find a solution using Frobenius method, that is: $$y_1(x) = ...
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Analytical solutions of Thomas Fermi equation

The Thomas Fermi model of atoms and nuclei is used in many applications of atomic and nuclear physics. The ODE related to this model is: $$\frac{d^2}{dx^2}\phi(x)=x^{-\frac{1}{2}}\phi(x)^{3/2}$$ with ...
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25 views

List of IVP known to have periodic solutions

I am looking for a list or review article describing differential equations and corresponding initial conditions which result in periodic solutions.
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64 views

Laplace Trouble to find solution

Trying to figure out how to use Laplace Transform to find $y(t)$: The problem is $$y''+4y'+4y=f(t)$$ where $f(t) = \cos(\omega t)$ if $0 < t < \pi$ and $f(t)=0$ if $t > \pi$? Initial ...
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What does it mean to say a differential equation is an eigenvalue problem?

My text says the following $$ \frac{\mathrm d}{\mathrm dx}\left(x^2 \frac{\mathrm dy}{\mathrm dx}\right) + \lambda y = 0,\;\;\;0\le x\le 1,\; y(1)=c\ge0$$ is an "eigenvalue problem". I don't ...
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42 views

Drawing phase portrait

This is the question in my textbook. I am a bit lost for 3 hours now. Could anyone please point me to the right direction? Let the $2 \times 2$ matrix $A$ have real, distinct eigenvalues $\lambda$ ...
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49 views

Solution to the differential equation $\frac{1}{2}\dot K-K^2+K=0$?

The solution the differential equation $\frac{1}{2}\dot K-K^2+K=0$ is given in the picture below Picture My solution $\frac{\mathrm{d} K}{2(K^2-K)}=\mathrm{d} t$ and ...
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Interior Products

Over on the Wiki page for interior products: http://en.wikipedia.org/wiki/Interior_product There is a line that says $\iota_X \alpha = \alpha(X) = \langle \alpha,X \rangle$ where $\alpha$ is a ...
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Differential vs difference equations in mathematical modeling

I'm reading a little about mathematical modeling and I've seen some population models based on differential equations. I've also seen some (not many) that can support both difference and differential ...
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Prove that a function is locally Lipschitz

I am studying the paper "F. D. Araruna, P. Braz E Silva, E. Zuazua, Asymptotic limits and stabilization for the 1D nonlinear Mindlin-Timoshenko ...
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76 views

Periodic Solution of Riccati Equation

I want to know for which condition on "$a$" and "$k$", i.e. for which function of $a(k)$, the following Riccati equation, with the initial condition $u(0)=ia$ ($i^2=-1$), have periodic solution with ...
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Inverse Laplace Transform of $\frac{2s+5}{s^2+4s+13} $ (Check My Solution)

I have solved, just need someone to check my solution is correct. My answer is - $$2e^{-2t}\cos(3t) + \dfrac{1}{3} e^{-2t}\sin(3t)$$ Thanks
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18 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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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 ...
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32 views

Existence and uniqueness of initial value problem in differential equation

consider the following equation: $$ y'=y^{\frac{1}{3}}, \,y(0)=0 $$ My question is how can I prove the existence and uniqueness of solutions of this initial value problem without solving the ...
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51 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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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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32 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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40 views

Solve by separating variables

$$\frac{dy}{dt}=e^y +1$$ I've tried: $$dy/dt - e^y = 1 $$ $$\Leftrightarrow y' - e^y dt = 1 dt$$ But I'm not sure what to do next or if I'm even doing this right!
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Homogeneous second-order differential equation with constant Wronskian

Problem Prove that if the Wronskian of an two solutions of differential equation $y''+p(x)y'+q(x)y=0$ is constant, then $p(x)$ is zero. My attempt. : Let $y_1$ and $y_2$ are solutions of given ...
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How to maximize speed of rest position approach of nonlinearly damped spring oscillator?

Inspired by comments to answer for this question: Suppose we have a system which is described by the equation $$\ddot x=-x+g(\dot x),$$ with initial conditions $x(0)=1$, $\dot x(0)=0$. If ...
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Proving the existence of a periodic solution

For a particular homework problem, I need to show that the differential equation: \begin{equation*} y^{\prime\prime}(x) - \frac{1-(y^{\prime}(x))^{2}}{1+(y^{\prime}(x))^{2}}y^{\prime} + y(x) = 0 ...
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Solution regarding Power Series and ODE's

About 4 months ago I posted Series solution to $y''-xy'-y=0$. I ran through the analysis and it appeared that I solved the ODE . The solution seemed to be ...
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59 views

Second order nonlinear ordinary differential equation. Help please

Can someone help me with this differential equation $$ay''(t)y(t)+2y'(t)=\left(b+\frac{c}{t^2}\right)y(t)^2$$
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non-smooth minmal surfaces and differenteial equations

This is the equation for a function $u(x,y)$ whose graph is a minimal surface (its mean curvature is $0$): $$(1+u_x^2)u_{yy}-2u_xu_yu_{xy}+(1+u_y^2)u_{xx}=0$$ My question is if there are non-smooth ...
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Matrix differential equation MX' = AX+XB+C(t)

Here is matrix differential equation: $$ \mu \frac {dX}{dt}=AX+XB+C(t) $$ $$ X(0) = X_0 $$ Here $\mu$ is real diagonal matrix, $X$ is $m$ by $n$ matrix. $A$, $B$ are real square matrices of constant ...
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39 views

How to prove Lyapounov stability of a circle orbit?

I am trying to go through exercises in V.Arnold's book on Mathematical methods of classical mechanics. There is a following exercise there: one considers a movement in a system with central potential ...
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Solution to the “cubic” Helmholtz equation

What is known about the solutions of the differential equation in three-dimensions $$ \nabla^2 \phi = -\kappa^2 (\phi + (1/3!)\phi^3) $$ Without the cubic term, this gives a linear operator ...
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45 views

Imaginary part of Log laplacian

I'm confused about how to calculate $\nabla^2 \log z$, where $z=re^{i\theta}$ is a complex number. My calculations return $$ \nabla^2 \log z = 2\pi\frac{\delta(r)}{r} [\delta(\theta) + i ...
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What are the different solution concepts for Matrix-Ordinary Differential Equation [Theory Question]

I was recently given a ODE to solve from a boss at work, with the knowledge that I haven't done them before and this will help me learn. I've spent 10 hours so far learning the basics of ODEs. The ...
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54 views

first order differential equation, proof

Prove that any solution $x(t)$ of the following differential equation: $$ \dot{x}+a(t)x=f(t), $$ where $ a(t)\geqslant c > 0 $ and $f(t) \rightarrow 0$ as $t \rightarrow \infty$, tends to 0 as ...