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

learn more… | top users | synonyms (1)

0
votes
0answers
38 views

eigenfunctions in a Sturm-Liouville problem

I've found that the eigenfunctions in a certain Sturm-Liouville problem satisfy a differential equation whose general solution is $\phi(x)= x^{a}[C_1M(a,2a+2,x)+C_2U(a,2a+2,x)]$, $x\ge0$, where $M$ is ...
0
votes
0answers
95 views

Boundary value problem: unique and multiple solution(s)

I have the following first order diff. equation: \begin{equation} \displaystyle i\frac{dy(x)}{dx} - \lambda y(x) = h(x) \end{equation} with periodic boundary conditions $y(0) = \pm y(-i\alpha)$, h(x) ...
0
votes
0answers
35 views

Eigenvalues and eigenvectors of a nonlinear operator

I have found a few nice answers to the question: "Why are eigenvalues and eigenvectors useful." I can imagine that knowledge of eigenvectors (-values) for a general nonlinear operator is worthless. ...
0
votes
0answers
75 views

Green function and Fourier series

I would like to understand what is wrong with the following computation. Indeed the result appears rather trivial. I have the following differential equation $$ y''(t,t')+z(t)y(t,t')=\delta(t-t') ...
0
votes
0answers
27 views

Solving differential functional equations with a restricted solution

I have a variable vector $X=\{x_1,x_2,...,x_n\}$, and a constant vector $V=\{v_1,v_2,...,v_n\}$. $f(x_i,X)$ is a function that takes X and xi as the parameter, for example: $f(x_i,X) = ...
0
votes
0answers
26 views

Implicit and Explicit formulation of this PDE

I am trying to solve the following equation by using both implicit as well as explicit formulation. $$\frac{\partial^2Q_i}{\partial t^2} = ks_{ij} \left( \frac{Q_j}{Ad_j} - \frac{Q_i}{Ad_i} \right) ...
0
votes
0answers
41 views

Simple Question- Is this a Sturm Liouville regular problem?

I have the following differential system: \begin{align} &(1-t^2)x'' -2tx' +\lambda x= 0 \\&x(0)=0 \\&x(l)+x'(l)=0 \end{align} I have to decide if it is an homogeneous regular Sturm ...
0
votes
0answers
95 views

SIR models - help!

I am doing a project on the spread of infectious diseases and I have chosen to model this spread by using the SIR model - (susceptibles, infectious, recovered). I have to insert data into this model ...
0
votes
0answers
28 views

Show that when $\epsilon $ < 0, the basin of attraction of the origin contains the region $z > −1$.

Consider the system $$x' = (\epsilon x+2y)(z+1)$$ $$y' = (-x+\epsilon y)(z+1)$$ $$ z' = -z^3$$ Show that when $\epsilon $ < 0, the basin of attraction of the origin contains the region $z > ...
0
votes
0answers
24 views

Proper equality sign for boundary value definition

Say I have a function that needs to define a boundary condition, like $f(0) = A$. In this case it is usually fairly self evident from context, that this is a requirement that $f(0)$ needs to satisfy. ...
0
votes
0answers
24 views

solve velocity equation with slop effect?

$$\frac{dp_l}{dx}-\mu_l\frac{\partial^2 u}{\partial y^2}=0$$ where $\mu_l$ and $p_l$ is the liquid phase viscosity and pressure, respectively; and $u$ is the flow velocity. The boundary ...
0
votes
0answers
23 views

Linear Differential Equation with Quasiperiodic Coefficient: A Question

I am reading the book Synergetics. The following passage is from the book: Assume that in the differential equation $\dot{q}= a(t)q$, where $a(t)$ is quasiperiodic, i. e., we assume that $a(t)$ ...
0
votes
0answers
95 views

Existence and uniqueness on this semi-linear parabolic PDE

I want to know whether the existence and uniqueness of a classical solution can be found about this question: Find a classical solution $u : [0,T]\times [0,\infty] \rightarrow {\mathbb R}$, such ...
0
votes
0answers
164 views

Euler-Lagrange Calculus of Variations Example

I have been working on solving Euler-Lagrange Equation problems in attempts to learn Calculus of Variations, but this one example has me stuck. I am probably making mistakes in my integration. I am ...
0
votes
0answers
38 views

Find a nonhyperbolic matrix which satisfies certain conditions

A nonhyperbolic matrix A for which the planar system $\overrightarrow x' = A \overrightarrow x$ has an equilibrium point at (0,0) with the x-axis as a stable curve and the y-axis as an unstable curve. ...
0
votes
0answers
73 views

Coupled mass spring system with damping and initial values

After researching through the web, I can't figure out how to express into a differential equation a coupled mass spring system with damping and initial values. Two masses and two springs, no external ...
0
votes
0answers
37 views

Frobenius' method help

I am trying to solve the following differential equation: $$y''-2xy'+2ny=0$$ using Frobenius' method. I hence substituted the following solution into the equation: $$y=\sum_{m=0}^\infty ...
0
votes
0answers
51 views

How to find the fundamental matrix for this differential equation?

I have the following differential equation: $$t^2x''(t)-4tx'(t)+6x(t)=0$$ and two linearly independent solutions, $x_1(t)=t^2$ and $x_2(t)=t^3$ in the interval $(0, \infty )$. I'm then asked to find ...
0
votes
0answers
48 views

Approximating a Markov process by differential equations

I have a system of states, $m_S = 1, 0, -1$. After performing a certain manipulation (it can be assumed to be instantaneous), a transition can happen with probability p. However, not all states can ...
0
votes
0answers
10 views

How to check that the vector fields of the system is tranversal to the boundary?

the system : and the feasible region : .please help, thank you for your attention :)
0
votes
0answers
43 views

Method of undetermined coefficients for finite difference approximations

I'm reading over my text, and the first mention of deriving the coefficients states: "Suppose we want a one-sided approximation to $u'(x)$ based on $u(x), u(x-h)$, and $u(x-2h)$ of the form: ...
0
votes
0answers
149 views

How many terms to use in a Taylor series for local truncation error

So from my understanding for a finite difference approximation, you're supposed to expand the series "about" the point $x$, e.g., $$u(x+h) = u(x) + h \ u'(x) + ...
0
votes
0answers
75 views

Proof the first-order equation

$$ \begin{align} \tau\dot y + y &= KF(t) \tag{1}\\ y(t) &= C_0 + C_1 e^{-t/\tau} \tag{2} \end{align} $$ How these two equations can form $$y(t)=KA+(y_0-KA)e^{-t/\tau}?$$ Note ...
0
votes
0answers
47 views

Stability of the ``square root'' differential equation

Suppose the system $$\dot{x}_i = f_i(x), ~~~ i = 1, \ldots, n ~~~~~~~~~~~ (1)$$ has the property that every trajectory which begins in the nonnegative orthant stays there and furthermore ...
0
votes
0answers
55 views

Solution of an Ordinary Differential Equation

I'd like to know if there is an analytical solution of the following Ordinary Differential Equation : $\displaystyle{\dfrac{d}{dt}x_i(t) = D_i + \sum_{j=1}^{n}L_{ij}x_j(t) + ...
0
votes
0answers
33 views

operator onto-theorem

I have this theorem: Let $V$ a Banach space, reflexive,separable, and let $A$ an operator monotonic, bounded, semi-continuos, coercive. Then, $A$ is onto. Where we can find the proof of this ...
0
votes
0answers
46 views

Sturm-Liouville boundary value problem with two different eigenfunctions

I am trying to express a function $f(x)$ in terms of a complete set of eigenfunctions found from a Sturm-Liouville boundary value problem: $$2y''(x)+4y'(x)+\lambda y(x)=0$$ $$y(0)=0, y'(2)=0$$ For ...
0
votes
0answers
41 views

A Sturm-Liouville problem

I've got the following Sturm-Liouville problem: $$ C x^2 y'' + y' = \lambda y, $$ with $C>0$, and initial conditions $$ e^{-C/x}y'=0 $$ at $x=0$ and at $x=+\infty$. I suspect that in this case the ...
0
votes
0answers
32 views

The integral $\int_A^B{FdR}$ is independent from the path $\Leftrightarrow$ $F=\bigtriangledown f$

Let $F$ be a vector field, $F=M\hat{i}+N\hat{j}+P\hat{k}$, where $M,N,P$:continuous at a region $D$. The integral $\int_A^B{FdR}$ is independent from the path from $A$ to $B$ in $D$ $\Leftrightarrow$ ...
0
votes
0answers
60 views

How to solve three second-order coupled PDE?

I need to solve these three second-order coupled partial differential equations: \begin{align} \left( A + B\frac{\partial^2}{\partial x^2} + C \left( \frac{\partial^2}{\partial y^2} + ...
0
votes
0answers
23 views

Uniform Stability on an Interval

I am trying to prove the following: Supposed $\phi (t)$ is a solution of $\dot{x}=f(t,x)$ defined on $(\alpha , \infty )$ and supposed $\alpha < \beta < \gamma$. They $\phi (t)$ is uniformly ...
0
votes
0answers
33 views

what do the higher differentials act graphically?

I wanna know what do actually the second order and higher order differentials mean. what are their geometrical interpretation, ya of course we solve them , use them but how we show them graphically, ...
0
votes
0answers
32 views

random walk differential equation

Was reading about random walks and I am learning differential equations and I thought combining them. Because I am just learning about these topics and want to know more, I will use the simplest ...
0
votes
0answers
35 views

Change of Variables in an Ordinary Differential Equation

Suppose we have an ordinary differential equation $$\tag{1} \frac{d^2 f}{dx^2}+a(x)\frac{d f}{dx}+b(x)f(x)=c(x)$$ It could potentially be useful to transform this differential equation into $$\tag{2} ...
0
votes
0answers
64 views

Local truncation error

(a) Find the local truncation error for the Trapezoidal rule $$ Y_{n+1} -Y_n= h/2( F_{n+1} + 3F_n)$$ and hence find the order of the method. What do you expect would happen to the local errors ...
0
votes
0answers
33 views

Equating two systems of PDEs

I'm trying to relate two sets of a pair of PDEs, but massively struggling! They should be equivalent up to a linear transformation. Any help would be wonderful! First set: $$\frac{\partial ...
0
votes
0answers
25 views

Solving the D.E:$\bigg(\dfrac{1}{t}+\dfrac{1}{t^2}-\dfrac{y}{t^2+y^2}\bigg)dt+\bigg(ye^y+\dfrac{t}{t^2+y^2} \bigg)dy=0$

By trying to solve: $$\bigg(\dfrac{1}{t}+\dfrac{1}{t^2}-\dfrac{y}{t^2+y^2}\bigg)dt+\bigg(ye^y+\dfrac{t}{t^2+y^2} \bigg)dy=0$$ And checking if the D.E. is exact, we partial derivate: ...
0
votes
0answers
40 views

initial value problem using picards theorem

I don't know how to do this problem and it would be a great one to know so I could do the similar problems like this one. How would I able to solve this ? thanks in advance.... Initial value problem ...
0
votes
0answers
25 views

Tangent solutions of $x' = f\left(\frac{x}{t}\right)$

Can someone help me in the following exercice? Lef $f:\mathbb{R}\longrightarrow \mathbb{R} $ be of $C^1$ class and $r\in\mathbb{R}$ such that $f(r) = r$. Show that If $f'(r) < 1$, ...
0
votes
0answers
71 views

Solving a weird Diff equation…

Good day people I am modelling a water bottle rocket. Using the conservation of mass : $$-{\rho}vA + \frac{d}{dt}∫dM = 0 \tag{1}$$ Since the mass, O2 pressure, O2 volume and velocity change over ...
0
votes
0answers
24 views

Approximation Methods for DEs

This is related to Euler's equation, but I'm being posed with the question: Show that $y(t) = \int e ^ {-u^2}$ satisfies the initial value problem $dy/dt = e^{-t^2}$, $y(0) = 0$. Can anyone clue me ...
0
votes
0answers
60 views

Dependence On Initial Conditions and Parameters.

I'm having a hard time getting started with this problem. I'm not even sure if this can be done by computing some derivatives or what not or if I need to use a proof for this solution.
0
votes
0answers
70 views

Existence of global solution to a system of ordinary differential equations

In Evans's Partial Differential Equations, second edition, pp 401, in establishing the existence of solution to wave equation, the author uses the Galerkin method and constructs a sequence of ...
0
votes
0answers
11 views

expansion of this function is incomplete

in differential equation of higher order , to find particular integral for algebraic functions , we reduce the differential co-efficient to $ (a+D)^{-n}$ form . so what will be its expansion . sould ...
0
votes
0answers
37 views

Complex Non-liner First order ODE problem

Good day people I am modelling a "water bottle rocket" using basic Continuum Mechanics. I have found a equation describing the acceleration of the rocket. I need to integrate this function to find ...
0
votes
0answers
34 views

MatCont continuation data for use on other plotting softwares

I have been using MatCont for generating continuation figures for my model ODEs. Dissatisfied with the quality of figures on MATLAB, I want to use gnuplot for plotting of this continuation data. In ...
0
votes
0answers
12 views

Finding new state via Euler integration

I have a state; call it X = [x,y, $\theta$]. It's a pose. I know my linear velocity (v) and my angular velocity ($\omega$) (or at least decent approximations of them). I've been asked to find the new ...
0
votes
0answers
28 views

How to solve first order differential equation

it is been a while since I solved differential equations. Hope you can help me with this one: $w(t)=\frac{u'}{u}$. What is the type of this equation? Thank you!
0
votes
0answers
42 views

Is this simplification valid?

The question is pretty straight forward: $$ \cos(x) dx + \sin(x)dx = (\cos(x) + \sin(x))dx $$ Thanks.
0
votes
0answers
518 views

partial differential equation by Lagrange's method given different solution

Let $$z= ax^2 + by^2 …\tag{1}$$ Let $\displaystyle\frac{\partial z}{\partial x}= 2ax$ and let $p = 2ax$ or $a\displaystyle = \frac p{2x}$ Similarly we get $\displaystyle b= \frac q{2y}$ If we plug ...