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

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27 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
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) & ...
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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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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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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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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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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 ...
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0answers
28 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} ...
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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
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: ...
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
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 ...
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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
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 = ...
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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 ...
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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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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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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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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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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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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
65 views

Solving the differential equation $y' \tan y = \frac1x$

Express the differential equation $$\tan y\,\frac{dy}{dx}=\frac{1}{x}$$ in a form not involving $\frac{dy}{dx}$. I undersand the concept of a differential equation (though, as a student, I am ...
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0answers
40 views

Solve the initial value problem 0f $x'=f(x),\quad x(0)=y$ [on hold]

Solve the initial value problem $$x'=f(x),\qquad x(0)=y$$ for $$f(x)=(x^2,x+x^{-1})^T$$ Denote the solution by $u(t,y)$ and compute $$Ф(t,y)=\frac{du}{dy}(t,y)$$ Compute the derivative $Df(x)$ for ...
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3answers
117 views

Separable differentiable equations

Which of the following is a solution to the separable differentiable equation: $$\frac{dy}{dx}=\frac{xy}{\ln y }$$ $A.\ \displaystyle e^{|x|}$ $B.\ \displaystyle e^{\sqrt{\frac{x^2}2}}$ $C.\ ...
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1answer
27 views

Finding the general solution of an ODE in matrix form using integration

so I have done this other times, but this seems to be a tricky case... The teacher also used a method that it is not in the book to show us another way of doing it, but I do not understand it :/ So ...
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0answers
80 views

need an equation [on hold]

I need an equation and I'm not very math oriented/dont have a lot of time to think so i figured, why not outsource? I am programming a pokemon battle simulator game. After a pokemon takes a blow, ...
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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 ...
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0answers
40 views

Elliptical Coordinates PDE, wave equation and separation of variables

I need some help with this problem. I know how to use the method of separation of variables and that the constant lambda should give you trig functions with solutions at some interval of pi, which ...
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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, ...
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1answer
15 views

Reducible to Separable First Order Differential Equation Word Problem in Analytic Geometry 1.4-29

I completed near all problems om a differential equations text chapter on reducing non-separable first order differential equations to separable by using an appropriate substitution for example u = ...
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1answer
14 views

Reducible to Separable First Order Differential Equation Word Problem in Analytic Geometry 1.4-28

I completed near all problems of a differential equations text chapter on reducing non-separable first order differential equations to separable by using an appropriate substitution for example $u = ...
2
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2answers
74 views

Intuition behind convolution identity for Laplace transforms

Convolutions, relatively speaking, are fairly straightforward for simple systems (from an applied perspective), but I cannot, at all, find the intuition behind the Laplace identity for convolutions. ...
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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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1answer
25 views

Reduce the equation to a homogeneous equation by a change of variables

The equation is $$ (x+1)^2 y'= (x+y)^2 -(y-1)(x+1) $$ I've tried substituting for $z=x+y$, $z=x+1$, etc. but they don't seem to give me anything that's homogeneous after rearranging.
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2answers
44 views

Does local existence in every point imply global existence for an ODE?

Consider the following first order ODE: $y' = f(t,y)$ subject to $y(t_{0}) = y_{0}$. I would like to show that there exists a unique function $y(\cdot)$ that passes through ($t_{0}, y_{0}$) The ...
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1answer
40 views

Trying to find 2nd power series solution

For the equation $ xy'' + 2xy' + 6e^xy = 0 $, I need to find the first 3 nonzero terms in each of two linearly independent solutions about x=0. I changed this to the form of ...
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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 ...
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1answer
23 views

Application of Ito's Lemma to a process

There is a function $S(X)=(A+1/b X_t)^b$, where $A$ and $b$ are constant I'll need to show how to get $dS = \frac13 S^{1/3} dX^2 + S^{2/3} dX$ and determine the value of $b$
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2answers
41 views

Show Uniqueness of Solution for Boundary Value Problem

Let $G \subseteq R^n$ be a simple, connected and bounded region with smooth boundary and let $f : \overline G \to \mathbb R$, $g : \partial G \to \mathbb R$ be continuous. Show that the following ...
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0answers
23 views

On the ODE $x^{\prime\prime}(t)+a(t)f(x(t))=0$

Problem. Let $a,f \in C^0(\mathbb R)$ with $a \ge 1$, $f \ge 0$ and suppose $\int_0^{+\infty} f = +\infty$. Let now $x$ be a solution of the ODE $$ x^{\prime\prime}(t) + a(t)f(x(t))=0. $$ Let ...
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3answers
48 views

Why does my derivation of $\mathcal{L(\frac{f(t)}{t})}$ lead to a wrong answer?

I'm trying to prove that $$\mathcal{L(\frac{f(t)}{t})(s)} = \int_s^{\infty}\mathcal{L(f(t))}(u)du$$ Here's my attempt: $$\mathcal{L(\frac{f(t)}{t})}(s)=\int_{0}^{\infty} \frac{f(t)}{t}e^{-st}dt$$ ...
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0answers
25 views

Finding alternating series for Power series

Given data and conditions I have a power series, $PS(x) = \sum_{n=0}^\infty R_nx^n$. I have a infinite GP,something like G(x) = $\sum_{k=0}^\infty ax^k = \frac{a}{1-x} $ . Never take G(x),such ...
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0answers
28 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} = ...
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3answers
31 views

Linearize a first order differential equation

The system described by $x'=2x^2-8$ is linearized about the equilibrium point -2. What is the resulting linearized equation? Answer is $x'=-8x-16$. How? I have no idea how it went from the first ...
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}= ...