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

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0answers
26 views

ODE with multiple simple conditions $f'(x)=f(x)(Ax+D ) $

I have an ODE to solve . The main issue is,in addition to solving it I have to keep some conditions too in the solution of f(x).. I am bit confused regarding how to deal with it. Equation is given ...
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0answers
50 views

Cauchy problem in pde [on hold]

I think that this question is related to Cauchy problem in PDE. Please explain the solution explicitly? And please which books should I study for such type questions? Especially, the book contains ...
1
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0answers
21 views

some questions on the solution of the Dirichlet's problem in the unit disk

Dirichlet's problem in the unit disk is to construct the harmonic function from the given continuous function on the boundary circle. It is solved by the convolution with the Poisson kernel, and we ...
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2answers
351 views

Differential Equations and Newtons method

How can I approach this question? For problem one this is what I did: Given the DE, $p'(x) = p''(x) + (2\pi*\frac{f}{c})^2p(x) = 0$, and its solution, $p(x) = sin(kx)$, I substituted the things on ...
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0answers
9 views

Stable manifold for bidimensional nonlinear dynamic system with complex eigenvalues

Given a autonomous nonlinear dynamic system of the form $$f(x,y)=\begin{bmatrix} B_1 x + g_1(x,y) \\ B_2 y + g_2(x,y) \end{bmatrix}$$ with $B_i\in\mathbb{R}$ (bidimensional problem), with $g_i$ ...
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0answers
9 views

Unable to get a particular solution for a system of ODE equations with the method of undetermined coefficients

so I have solved this problem using another method (the method of diagonalisation), but I now want to try with the method of undetermined coefficients and cannot get the right result for $\vec{b}$. ...
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1answer
36 views

Solving the second order differential equation $d^2u /dt^2 =a$

Let $\frac{\partial^2 u}{\partial t^2}=a$ which $ a$ is constant, then $u=\frac{a}{2}t^2+bt+c$ on interval $ [0,T)$. Let's say we have $a, c$. Then how can we find $b$?
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1answer
36 views

First-order linear differential equation for matrix valued functions of size $3\times 3$

I have two matries given by (M' means derivative w.r.t x) $ M=\left( \begin{array}{ccc} f_1(x) & f_2(x) & f_3(x) \\ f_4(x) & f_5(x)& f_6(x) \\ f_7(x) & f_8(x) & ...
1
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2answers
37 views

Solve the following Ordinary Differential Equation

Suppose if we want to solve $\frac{d}{dx}u + f(x)u(x) = 0,$ then the solution is $u(x)=u(0)e^{-\int_0^xf(y)dy}$. Similarly what is the solution of $\frac{d^2}{dx^2}u- \frac{d}{dx}u - f(x)u(x) = 0,$ ...
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3answers
240 views

Solving inhomogenous ODE

I have an inhomogenous ODE. The main issue here is variables are matrices. It is bit of matrix calculus. A solution would be highly appreciated interms of x . I guess we can use same methods for ...
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0answers
40 views

Method of Characteristics for a PDE

I'm working through a problem at the moment, and I've got an answer, but it seems far too complicated... I've been asked to use the method of characteristics to solve the following PDE; $$x^2 u_x ...
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1answer
136 views

Solve a differential equation using the power series method

Problem By assuming a power series solution of the form $$y(x) = \sum_{m=0}^{\infty} c_mx^m , \quad c_0 \not =0 $$ Show that the equation $ 2y'+xy=x $ has general solution ...
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2answers
13 views

BVP Second order ODE Infinite number of solutions

I'm attempting to solve $X''(x)+ \lambda ^2X=0$ with the boundary conditions: $X(0)=0$ and $X(L)=0$. However I do not understand why we have $B_n$ Is this something to do with the linearity of the ...
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1answer
49 views

Differential equation how to find out bounded or not bounded

For which value of the parameter $k$ will all solutions of remain bounded as $t \rightarrow \infty$? $$u''(t)+k~u(t)=2 \sin(10t)?$$
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0answers
21 views

ODE Initial value problem formualtion

If I have the following ODE initial value problem, $$\begin{align} y'(t) &= f(t), \quad t>0, \\ y(0) &= y_0. \end{align}$$ Then I was taught that a solution to the problem is given by ...
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0answers
20 views

PDE and Taylor's formula

I'm looking to a prove that a function that satisfies the following equations is actually $f(x,t)=x^3 \pm tx$ after changing coordinates. Here are the equations: 1) $\frac{\partial^3 f}{\partial ...
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0answers
10 views

ODE theorem solution existence

My memories from ODEs are a little vague. I need a theorem that would explain the following: If $\phi_t$ is a family of functions defined on $R^n$ with values in $R^n$ such that ...
5
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1answer
154 views
+50

Uniform continuity of the function $x(t)=e^{tA}x$

Let $A$ be a bounded operator on a Banach space $X$. Consider the exponential function $x(t)=e^{tA}x:=\sum_{n=0}^{+\infty}\dfrac{t^nA^n}{n!}x$, for all $t\in \mathbb{R}$, where $x\in X$. If the ...
1
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2answers
534 views

Symbol of a (non linear) differential operator

I am interested in knowing whether there is a definition for the symbol of a PDO which is NOT linear. In Wikipedia and in the book I am reading (An Introduction to Partial Differential Equations by ...
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2answers
336 views

The trace-determinant plane, classification of equilibria of differential equations

What are some easy ways to remember each of the different behaviors of general solutions of ordinary differential equations in the trace-determinant plane? For differential equations of the form ...
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1answer
23 views

Stability of a linear time-varying system

I have got the following system: $\dot{z}_2 = - \gamma_2 \left ( \begin{bmatrix} \sin^2(x_1(t)) & \sin(x_1(t))x_2(t)\\ x_2(t)\sin(x_1(t)) & x_2(t)^2 \end{bmatrix} \right ) z_2$ which ...
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0answers
27 views

Proving that a Sturm-Liouville problem is in the limit-point/-circle case

I would like to understand techniques anybody is able to detail to me on how one may actually prove that a particular Sturm-Liouville (S-L) problem, i.e., of the form \begin{equation} ...
3
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1answer
42 views

Changes of variables to get an Elliptic Integral of the First Kind

I'm working with a non-linear second order ODE which has an analytical solution in terms of the Jacobi elliptical function $sn(u|k^2)$. The equation is $y''=y(\gamma - \frac{y^2}{2})$ where $\gamma$ ...
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1answer
32 views

Solve the given differential equation by using Green's function method

I am really struggling with the concept and handling of the green's function. I have to solve the given differential equation using Green's function method $\frac{d^{2}y}{dx^{2}}+k^{2}y=\delta ...
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1answer
49 views

finding an explicit formula from arctangent

Seems straight-forward but i can't get it right; I have this implicit equation: $$-2 \arctan{ \left( \frac{\sqrt{y-y^2}}{y} \right) } -x=c$$ where c is a constant. I've to find the explicit ...
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0answers
20 views

How to solve a 4x4 linear system using a set of initial conditions

I have a non linear 4x4 system of ODEs. I linearized it about an equilibrium point (i am using floquet theory and i need the linearized system for that) and below is the linearized system: ...
1
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1answer
37 views

Solution to second order differential equation

I'm reading a paper in which the authors solve the following equation: $\frac{d^{2}}{dz^{2}}\hat{p}$($\bf{q}$$,z)$-$q^{2}\hat{p}$($\bf{q}$$,z)$-$\frac{iq_{y}}{(2\pi)^{2}}\delta(z-z_{2})$=0 here ...
3
votes
4answers
374 views

If $f$ is holomorphic and $\,f'' = f$, then $f(z) = A \cosh z + B \sinh z$

Suppose $f$ is holomorphic in a disk centered at the origin and $f$ satisfies the differential equation $$f'' = f.$$ Show that $f$ is of the form $$f(z)=A \sinh z + B \cosh z,$$ for suitable constants ...
3
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2answers
77 views

Solving a particular system of differential equations

The problem I'm trying to solve is this: $X'(t) \in \mathbb{R}^3 \,, \, \omega = (\omega_1,\omega_2,\omega_3) $ Find the general solution for $$X'(t) = \omega \times X(t)$$ After doing the cross ...
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1answer
55 views

Existence of solution of ordinary differential equation

I am reading a proof of the existence of solutions for ordinary differential equations and I have some basic doubt. I'll copy the statement, the part of the proof I don't understand and my question: ...
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0answers
20 views

Solving a simple Recurrence in summation form(very special case)

I have a bit confusing recursion form $\sum_{n=2}^{\infty}\{f(n)\frac{n}{n-1}\}=C, \tag 1$ $f(0)=b,f(1)= a,f(2)=c$ and $C$ are constants. Could you help me to solve this recursion or help me to ...
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2answers
24 views

Finding an ODE with specified solutions

So I have this math problem: Find functions $p(x)$ and $q(x)$ so that $y_1(x)=\sin x$ and $y_2=x*\sin x$ are solutions of the differential equation $y''+p(x)y'+q(x)y=0$. I'm just so lost as to what ...
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3answers
573 views

Separable differential equation modeling salt dissolved in water

I have no idea what to do for this. The equation is supposed to model a solution having salty water dumped into a tank that leaks the solution. ? A tank contains 1000L of pure water. Brine that ...
9
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4answers
161 views

Solution to $y(x) + y'(x) + y''(x) + y'''(x) + \cdots = 0$

Is there a non-trivial solution to the following differential equation? $$y(x) + y'(x) + y''(x) + y'''(x) + \cdots= 0$$ That is, is there a smooth function $y : \mathbb{R} \to \mathbb{R}$ such that ...
1
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2answers
45 views

Exponential Growth Differential Equation

A population of buffalo grows exponentially (the rate of growth is determined by the population itself) but has a carrying capacity. Its population (in tens of thousands) at a time t ( in years ) is ...
3
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3answers
48 views

Differential Equation $\frac{dP}{dt} = kP(1-P)$

I have a question about solving this differential equation. So, the question is to solve it given that $P(0)=\frac23$ So this is what I've done so far $$\frac{dP}{dt} = kP(1-P)$$ $$ k\,dt = ...
2
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1answer
76 views

Differential Equation $\frac{dy}{dt}$ = $y - t$

Given the differential equation $\dfrac{dy}{dt}$ = $y - t$ Is this equation separable? -> No it is impossible to separate this equation because we can't get $y$ alone with $dy$ and $-t$ alone with ...
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0answers
13 views

Initial value problem test cases

I am working on some materials about numerical solution of initial value problem for ODEs. Are there any state of art test cases used to test properties of methods? I have found one in Wikipedia and ...
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0answers
47 views

Long-time asymptotic behaviour of a system of two ODEs

We have the following nonlinear ODE: $$ f' = af-bg -(f+g)^k \bigl(f'(0) +g'(0)\bigr) + f'(0), $$ $$ \bigl(G-T(x)\bigr) g' = -af+bg - g'(0), $$ where $a,b,k,G$ are constants, $f'(0)$ and $g'(0)$ are ...
0
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1answer
72 views

Solving for Center Manifold with Parameter

I have a system of ODEs given by $$\frac{dX}{d\tau}=\beta X\left(1 - \frac{X+Y}{N}\right)$$ $$\frac{dY}{d\tau}= Y\left(1 - \frac{X+Y}{N}\right)$$ where $\beta $ is a parameter. How should I ...
5
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2answers
73 views

System of non-linear ODE's

do you have any suggestions to solve analytically the Non-linear ODE system $\dot x=18 x^2 y-3p x^2+6p xy$ $\dot y=18 x^2 y-6p xy $ where $p$ is a real constant. Thank you very much cheers
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0answers
24 views

existence and uniqueness of volterra integral equation of the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
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2answers
53 views

How to find the intersection points of lines that are normal to two curves?

Let I have two curves, \begin{gather} f(x)=\frac{x^3}{4}+1 \\ g(x)=\frac{(x-\tfrac{1}{2})^3}{7}+\tfrac{1}{2} \end{gather} There are zero or more lines that are normal to both curves. In other words, ...
2
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1answer
64 views
+50

Asymptotic expansion on 3 nonlinear ordinary differential equations

The 3 nonlinear differential equations are as follows \begin{equation} \epsilon \frac{dc}{dt}=\alpha I + \ c (-K_F - K_D-K_N s-K_P(1-q)), \nonumber \end{equation} \begin{equation} \frac{ds}{dt}= ...
4
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2answers
126 views

Solve nonlinear differential equation

Could you help me solve or give me some advice about following differential equation $$ 2(y')^2 + 3xy'y'' + 3yy'' = 0 $$
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2answers
28 views

The set consisting of all solutions of a homogeneous linear differential equation of order n is a vector space.

The set $S$ consisting of all solutions of a homogeneous linear differential equation of order $n$ is a vector space.
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0answers
45 views

On a differential equation problem of international mathematical competition for university students

I am trying to solve problem 2 of this competition: http://www.imc-math.org.uk/imc2009/imc2009-day2-solutions.pdf I have other thought but i couldn't fill in the detail. Consider the initial value ...
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1answer
52 views

Initial values are lost (diff eq to Transfer function)?

I read eternal Julius O. Smith III and he says that $$x_{n-m} = z^{-m}X(z)$$ Particularly, difference relation $$y_{n} = y_{n-1} + x_{n}$$ is solved by by $$Y = z^{-1}Y + X = {X \over ...
2
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1answer
35 views

Prerequisite of Dynamical system and applied PDE

For the further research interest, I want to focus on the application of Dynamical systems and PDE in the field of robotics and neuroscience, particularly from a mathematical points of view. ...
1
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3answers
50 views

Inverse Laplace Transform,

I have been stuck on this problem for quite a bit, have tried to look at similar answers on website but no help... The original questions is, Solve the IVP $\ y''+y=\sin(t);y(0)=1;y'(0)=0$ I ...