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

learn more… | top users | synonyms (1)

2
votes
0answers
41 views

Green's function for Dirichlet Laplacian

I am thinking of the Dirichlet boundary condition $u|_{\partial \Omega}$ for a domain $\Omega \subset \mathbb{R}^n$. Let $\Delta$ be the Dirichlet Laplacian, which accepts only functions with the ...
2
votes
0answers
36 views

finding solution to a partial integro differential equation

I want to find a function (or a set of functions) such that $u(x,t)$ satisfies the following partial integro-differential equation with singular kernel \begin{eqnarray} &&u_x(0,t) = \int_0^t ...
2
votes
0answers
52 views

Combining two differential equations

I have two differential equations that are connected by an equation, $L_1\frac{d^2I_1}{dt^2} + \frac{1}{C_1}I_1=\frac{dV}{dt}$ $L_2\frac{d^2I_2}{dt^2} + \frac{1}{C_2}I_2=\frac{dV}{dt}$ $I_1+I_2=I$ ...
2
votes
0answers
16 views

Existenence of the solution for a PDE-ODE system.

I have the PDE-ODE system below: $\frac{\partial c}{\partial t}= D \Delta c - \eta \nabla.(c\nabla v)+g(c,v)$ $\frac{dv}{dt}=-\alpha cv+\xi(c,v)$ with initial conditions and Neumann boundary ...
2
votes
0answers
20 views

Proof that maximal interval of existence exist and bounded

For each $\lambda\in \mathbb{R}$, let $\varphi_{\lambda}$ : $J_{\lambda}\rightarrow \mathbb{R}$ denote the solution to the following initial value problem: $$ ...
2
votes
0answers
47 views

How to solve second order differential equations? [summary]

As I do my engineering studies, I find more and more ways to solve differential equations, especially the second order ones. With more and more ways to solve these equations, I am loosing my overview ...
2
votes
0answers
24 views

How do I approximate $f''(x)+(E-U(x))f(x)=0$ for a piecewise $U$ and find $E$?

I am trying to approximate the solution to the equation $f''(x)+(E-U(x))f(x)=0$ where $U(x) = \begin{cases} \frac{U_0}{m}x-U_0 & \text{for $-m<x<0$} \\ \frac{-U_0}{m}x-U_0 & ...
2
votes
0answers
29 views

Find a series solution to $(x^2-2)y''+6xy'+4y=0$.

Find a series solution to $(x^2-2)y''+6xy'+4y=0$. A. Find the recurrence relation to $a_n$: My answer is $a_{n+2}=a_n\cdot \frac{n+4}{2(n+2)}$ which is correct. B. Using A, write two independent ...
2
votes
0answers
38 views

Solution of $\Pi(y(x)+1)+\sin(x)=y(x)+y'(x)$

How do we solve $$\Pi(y(x)+1)+\sin(x)=y(x)+y'(x)$$ I suspect it will be a function of many cases. The solution of $$\Pi(x+1)+\sin(x)=y(x)+y'(x)$$ is hard only at the evaluation of the last integral ...
2
votes
0answers
28 views

The existence of integral equations solution for a 2-dimensional unknown function

Suppose $f(\cdot,\cdot)\in C[0,1]^2$ is a kernel. $f$ is integratble $\int_0^1\int_0^1 f(x,y)dxdy<\infty$. $a,b\in C[0,1]$ are known functions, and $z(\cdot,\cdot)\in C[0,1]^2$ is a 2-dimensional ...
2
votes
0answers
59 views

Forced oscillation in a pendulum and resonances

In a pendulum without the small angles approximation the equation describing the motion of the mass is: $$\ddot{\phi}(t)=-\dfrac{g}{l}\sin\left(\phi(t)\right)$$ Applying a sinusoidal force ...
2
votes
0answers
35 views

Taking partial derivatives over multiple summations

I have the following equation obtained from one of the models. $\mathcal{H} = \sum\limits_{D} \sum\limits_{W}n(d,w)\sum\limits_{Z} p(z|d,w)[\log{p(d)}+\log{p(z|d)}+\log{p(w|z)]}$ I need to take ...
2
votes
0answers
28 views

Solution of partial difference equation

I want to find the explicit solution of the following difference equation $e_{i,j+1}=re_{i-1,j}+(1-2r)e_{i,j}+re_{i+1,j}+km_{i,j}$ where $r>0$, $k>0$ and $m_{i,j}$ are known and $e_{i,0}=0$. ...
2
votes
0answers
35 views

Is ODE theory useful for developing numerical solvers for ODEs?

I will be doing research in developing numerical solvers for ODEs. I was wondering if knowledge of ODE theory will be useful and if so in what ways. I am asking because, I am inclined to take a ...
2
votes
0answers
47 views

Adding a delta function to a differential equation

So say I have a differential equation of the form: $$ \left(\alpha \frac{d^2}{dx^2}+fx^2 \right)y(x)=\lambda y(x) $$ Whose solutions are known (a Gaussian multiplying a Hermite polynomial.) I am now ...
2
votes
0answers
51 views

second order differential equation with Green's function

I need to solve following differential equation \begin{eqnarray} y''(x) - k = \delta(x-x_0) \end{eqnarray} subject to conditions: \begin{eqnarray} y(x=-a) = 0 \\ y(x=b) = p \end{eqnarray} Is it ...
2
votes
0answers
23 views

using ode45 for descriptive forms

Using Matlab what would be the most efficient way to solve, $A_1x'(t) = A_2x(t)$, where both $A_1$ and $A_2$ are $n\times n$ matrices. Both are sparse matrices and hence I want to avoid inversion. ...
2
votes
0answers
43 views

Finding a solution basis of differential equation

Find a solution basis of $$y'=\left[ \begin{matrix}3&-4&-2\\2&-3&-2\\0&0&1\\ \end{matrix} \right]y \,\text{ and find the solution } \Phi \text{ with } \Phi(0) = (1,1,1).$$ I'm ...
2
votes
0answers
124 views

Quaternion conversion

We have a normalized orthogonal co-ordinate frame travelling through the curve as in figure 1 below, from one end to other. Let us call starting end as A and ending end as B. What we know is initial ...
2
votes
0answers
33 views

Special properties of bounded functions

I have a problem understanding the reasons as to why under some circumstances a term can be omitted due to it being a part of a bounded function, and I hoped to get some clarity to this here. There is ...
2
votes
0answers
23 views

Growth condition in differential equation and vanishing solution at boundaries

In a discussion on solving a partial differential equation I lately read: "Under a standard growth condition on the solution at infinity, the resulting PDE is fully specified without boundary ...
2
votes
0answers
34 views

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} = ...
2
votes
0answers
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 ...
2
votes
0answers
66 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 ...
2
votes
0answers
37 views

How to solve the following an ODE?

Let $x,y,z$ be a given point in $\mathbb{R}^3$. How to solve $(x'(t),y'(t),z'(t))=(x(x+y+z), y(x+y+z),z(x+y+z))$?
2
votes
0answers
132 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 ...
2
votes
0answers
55 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 ...
2
votes
0answers
104 views

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 ...
2
votes
0answers
65 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} ...
2
votes
0answers
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
2
votes
0answers
75 views

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 ...
2
votes
0answers
48 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 ...
2
votes
0answers
160 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, ...
2
votes
0answers
47 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 ...
2
votes
0answers
170 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 ...
2
votes
0answers
38 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 ...
2
votes
0answers
34 views

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)\\ ...
2
votes
0answers
53 views

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' = ...
2
votes
0answers
66 views

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 ...
2
votes
0answers
26 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.
2
votes
0answers
68 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 ...
2
votes
0answers
42 views

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 ...
2
votes
0answers
51 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 $\lambda$ ...
2
votes
0answers
50 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 ...
2
votes
0answers
36 views

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 ...
2
votes
0answers
190 views

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 ...
2
votes
0answers
60 views

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 ...
2
votes
0answers
86 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 ...
2
votes
0answers
27 views

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
2
votes
0answers
25 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 $$\mathbf{x}(t)=(x_1(t),\ldots,x_n(t)),$$ ...