Questions tagged [ordinary-differential-equations]

For questions about ordinary differential equations, which are differential equations involving ordinary derivatives of one or more dependent variables with respect to a single independent variables. For questions specifically concerning partial differential equations, use the [tag:pde] instead.

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13 views

Uniqueness of the Solution of a Ordinary Differential Equation

Let $f$ a function, and the Cauchy problem $$x'=f(t,x) \qquad x(t_0)=x_0$$ I am studying EDO from two books, and I have a question about the uniqueness of the solution. Book 1: If $f$ is $C^1$ ...
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find the solution of $2x\sin{\left(\frac{y}{x}\right)}dx+3y\cos{\left(\frac{y}{x}\right)}dx-3x\cos{\left(\frac{y}{x}\right)}dy=0$

A solution of the equation $$2x\sin{\left(\frac{y}{x}\right)}dx+3y\cos{\left(\frac{y}{x}\right)}dx-3x\cos{\left(\frac{y}{x}\right)}dy=0$$ I know the answer $c\sin(3y/x)$ but I don't know the solution
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35 views

Find the general solution of $y''+p(t)y'+q(t)y = 1+t$

A solution of the equation $$y''+p(t)y'+q(t)y = 0$$ is $1+t^2$, and the Wronskian of any of two solutions of the equation is constant. Find the general solution of $$y''+p(t)y'+q(t)y = 1+t$$ I have ...
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1answer
13 views

Property of solution to a Cauchy problem

Let $I\subseteq \Bbb R$ an interval, let $w$ differentiable on $I$ such that $$w'(t)\le L|w(t)|\qquad \forall t\in I$$ for some $L>0.$ Let $t_0\in I$. Prove that $$w(t_0)\le 0\implies w(t)\le0\quad ...
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15 views

Eigenvalue problem on the real line

The following is a problem in a text (in Portuguese) on Critical Point Theory that I am reading: Find the eigenvalues and eigenfunctions of the problem $$ (P) \quad \begin{cases} - y'' = \lambda ...
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Approach for non linear ODE with quadratic derivative.

What is the approach to solve the following nonlinear ODE: $$\ddot{x}(t) + a(\dot{x}(t))^2+b\dot{x}(t)+kx(t) = 0$$ where $a$ is positive when $\dot{x}<0$ and negative otherwise. I am trying to ...
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1answer
29 views

When (and why) does a linear vector field enjoy polynomial conserved quantities?

The familiar equations $\sin^2 + \cos^2 = 1$ and $\cosh^2 - \sinh^2 = 1$ can be explained by noticing that the corresponding 2-dimensional linear ODE have particularly simple conserved quantities; if $...
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1answer
21 views

transform a 2 dimensional ode 1 system to 2nd order one dimension system

Given a matrix $M$ of $2 \times 2$, and an ode: $$y'=My$$ let $y=(v_1,v_2)$. find a second order ode such $v_1,v_2$ are solutions.
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32 views

Property of solution to a differential equation

Let $a(t),b(t)$ continuous functions in $I$ closed and bounded interval in $\Bbb R$. Let $t_0\in I, y_0\in \Bbb R$ and $y(t)$ solution for $$ \left\{ \begin{array}{c} y'=a(t)y+b(t) \\ y(t_0)=y_0 \...
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11 views

Linearised Equation from Lubrication Theory

I am reading this article and am unclear as to how equation $(3.9)$ is obtained. One is supposed to go from $H''' = \frac{1-H}{H^3}$ to $\epsilon = A e^{-\zeta} + B \cos(\sqrt{3} \zeta /2 + \phi) ...
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1answer
32 views

Find the general solution for this equation.

Three solutions of a non-homogeneous second order linear equation are $$\psi _1(t)=1+e^{t^2}, \quad \psi _2(t)=1+te^{t^2} \quad \text{and} \quad \psi _3(t)=(t+1)e^{t^2}+1 $$ Find the general solution ...
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Proof that $\overline F(t) = (\overline v_1,\overline v_2)$ is the fundamental matrix of a linear differential system

I have a project where they ask me to prove that the matrix $\overline F$(t) = ($\overline v_1$,$\overline v_2$), where $\overline v_1$+i$\overline v_2$ = $e^{{λ_1}t}$$\overline r_1$ = $e^{rt}$(cos(ωt)...
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Differential Equation that do not contain “y” directly.

I've been on this question for two days and I can't seem to get past the last integral. The natural logarithm of y seems to be a problem $\mathrm{ y\frac{d^2y}{dx^2} + \frac{dy}{dx} + \Bigl(\frac{dy}{...
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2answers
75 views

Prove that $0$ is the only $2\pi$-periodic solution of $\ddot{x}+3x+x^3=0$

Prove that $0$ is the only $2\pi$-periodic solution of $\ddot{x}+3x+x^3=0$. I don't know how to deal with this non-linear differential equation. I tried to consider $\ddot{x}(t+2\pi)+3x(t+2\pi)+x^3(t+...
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2answers
24 views

Stability of the trivial solution of a system of differential equations

I am trying to determine the stability properties of the equilbrium solution $(x,y) = (0,0)$ of the following system of ODEs: $$ \dot x = x - y + kx(x^2+y^2), \\ \dot y = x - y + ky(x^2+y^2), $$ ...
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1answer
36 views

Specific differential equation

This is the differential equation: $$y'+{y \over {\sin(2t)}}=\sqrt {\sin(t)}$$ I found: $$\int{1\over \sin(2t)}dt= \ln\sqrt{|\tan(t)|}$$ So the next step is: $$\int{e^{\ln\sqrt{|\tan(t)|}}\sqrt{\sin(...
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Solution for Comparative Statistics with higher degree first order derivation…How to approach this?

My aim is it to obtain comparative statistics $\frac{\partial s}{\partial I_D}$ and $\frac{\partial s}{\partial I_R}$for following functional form. My maximisation problem in form of the Lagrangian: $...
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26 views

How to solve these nonlinear ODEs numerically without enough boundary conditions?

\begin{aligned} x_1'(t) &= x_2(t)\\ x_2'(t) &= -0.1 \left(100\ \text{sgn}(p_2(t))-10 x_1(t) x_4(t)^2+50 x_2(t)+98 \sin (x_3(t))\right)\\ x_3'(t) &= x_4(t)\\ x_4'(t) &= -\frac{0.1 (1000\...
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2answers
38 views

finding solution of differential equation $y''-9y=(x^2-2)\sin(4x)$

Using method of undetermined coefficient finding solution of $y''-9y=(x^2-2)\sin(4x)$ What i try: First we will find characteristic solution $r^2-9=0\Longrightarrow r=\pm 3$ So our characteristic ...
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1answer
16 views

Verification of power series solution to differential equation.

Verify that $$y=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}x^n$$ is the solution of the differential equation $(x+1)y+y’=0$. So we differentiate $y$ to get $$y’=\sum_{n=1}^{\infty} (-1)^{n+1}x^{n-1}$$ ...
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1answer
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Heat equation in spherical coordinates: heat flow at boundary

At steady-state we can write the one-dimensional heat equation as $$\frac{\partial}{\partial r}\left(r^2k\frac{\partial T}{\partial r}\right)=-r^2Q$$ which has the solution $$T(r)=-\frac{r^2Q}{6k}+\...
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1answer
26 views

Proof that the zero solution of equation $y''+f(y)=0$ is sustainable

I am trying to solve this prolem. Let $f(0) = 0$ and $tf(t)>0$ for $t \neq 0$. Proof that the zero solution of equation $$y''+f(y)=0$$ is sustainable. I thin it is a bad way to solve it by using ...
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1answer
30 views

SIR Model Specifics

I read on Wikipedia (under "Compartmental models in epidemiology") that the differential equations for the SIR Model was the following, $$S'(t)=-\frac{\beta}{N}I(t)S(t)$$ $$I'(t)=\frac{\beta}{N}I(t)S(...
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2answers
60 views

Integral Curves of Vector Fields with Zero Divergence or Zero Curl

Say we've got some vector field which at every point indicates the instantaneous velocity of a particle moving through that point. I'm trying to gain some intuition for what the possible ...
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1answer
29 views

SL-Eigenvalue/function problem with arbitrary boundary values

The problem is to find all eigenvalues and eigenfunctions for the following SL system. $u'' + \lambda u = 0, x \in [a,b]$ $u'(a) = u'(b) = 0$ I know the general idea of how to do these problems ...
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0answers
17 views

Literature request — uniqueness and existence of a specific type of ODE

I am looking for a proof of the existence and uniquenes of ODE's of the type: \begin{equation} \dot{f}(t,x,y) = F(h(t,x), f(t,x,y)), \end{equation} where $f : T \times X ...
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0answers
37 views

How to solve this relatively simple non-linear ODE?

I'm having troubles in simplifying a differential equation to find its solutions. Consider this ODE: $$ \frac{1}{r} \, \frac{d}{dr} \Bigl( \frac{r}{B} \, \frac{d B}{dr} \Bigr) = k \, B^2, \tag{1} ...
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23 views

First order Non-Linear ODE with no explicit form of the derivative

I have the following non-linear first order ODE $$ [a+b(1-e^{-m\frac{dy}{dx}})]\frac{dy}{dx}=f(x) $$ to be integrated over the range $x=x_0$ to $x=x_1$ The ODE is of the form $$ y'=f(x,y,y') $$ ...
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14 views

An explication about the derivate of poincaré maps

I am studying about derivate of poincaré maps $(\mathbb{R^2})$ but i don't know how to used this. I don't know how found the poincaré maps too. Someone can give me an easy exemple or open my mind in ...
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What is the solution of this dif eqn [closed]

enter image description here The question is not in english but it is clear. Asking the solution with given initial conditions
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48 views

How do you solve $(D^2+1)y = \ln|\cos(x)|$

I am stuch on this question: $$(D^2+1)y = \ln|\cos x| $$ where $D^2$ denotes the differential operator $d^2y/dx^2$ I suppose that I will begin with these lines: For cos(x) function use $(D^2+1)$...
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2answers
56 views

Central Difference Approximations

Hi Guys I was going through the different approximations which can be used for differentiation such as the forward difference, the backward difference and lastly the central difference approximations. ...
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35 views

Differential Equation using finite difference method

I am working on the following question $$y''+8(\sin^2 \pi y) y=0$$ where the initial conditions are $$y(0) = y(1) = 1$$ Now by the finite difference method i have made the substitution for $y''$ ...
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2answers
28 views

Trivial Solution in Differential Equation

in the following differential equation: $xy' + y = y^{-2} $ we can see that $y=1$ is a solution that always satisfies the equation regardless of the value of $x$. Do we call this a trivial ...
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13 views

Shooting Method for non-linear coupled ODE DAE

I have a 2 coupled non-linear ODEs that make up a Differential Algebraic Equation. $$V' = -\frac{V}{2\delta}\delta ' - \frac{6}{\delta ^2 } + \frac{6}{V}\Delta T$$ $$\delta ' = \frac{6}{\delta V}-\...
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3answers
36 views

Solve this differential equation $x^2y''-5xy'+6y=0$

Solve this equation $$ \begin{cases} x^2y''-5xy'+6y=0 \\ y(-1)=3 \\ y'(-1)=2 \end{cases} $$ I got $$y=c_1x^{3+\sqrt3}+c_2x^{3-\sqrt3}$$ I have three little questions. Could I solve the problem by ...
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19 views

Proof of existence and uniqueness of solution under control

I read a research paper in which a replicator dynamics ODE is considered: $\dot{x}_{i}(t)=\delta x_{i}(t)[\pi(i, \mathbf{x}(t), \mathbf{r}(t))-\pi(\mathbf{x}(t), \mathbf{x}(t), \mathbf{r}(t))]$ To ...
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1answer
27 views

General solution of $tx''-x'+4t^3x=4t^3$

The task is to find general solution of: $tx''-x'+4t^3x=4t^3$ The hint is to substitute $s=t^2$ My attempt: First I guessed that $x=1$ satisfies the equation and that is our particular solution. ...
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2answers
83 views

Estimate blow-up time.

How am I supposed to find the blow-up time of this ODE solution? $$y'=e^x + y^2 \qquad y(0)=0$$ The fact that it blows up it's granted by the fact that $y' \geq y^2$ which solution explodes. But how ...
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0answers
22 views

Solving $2 \alpha k=\omega^{2} \mu {\epsilon}^{''}+\omega \mu \sigma,k^{2}-\alpha^{2}=\omega^{2} \mu {\epsilon}^{'}$

I need help in this question please , i didn't understand it efficiently to solve it Solve for $k$ and $\alpha$ \begin{array}{l} 2 \alpha k=\omega^{2} \mu {\epsilon}^{''}+\omega \mu \sigma \\ k^{2}-...
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1answer
31 views

Second order ODE solution - help me spot a mistake

Solve following ODE: $$ (1-x)x''+2(x')^2=0; x(0)=2, x'(0)=-1 $$ $$ x''=\frac{-2(x')^2}{1-x} $$ substitute $x'=u(x)$ and assume $u \neq 0$ $$ uu'=\frac{-2(u)^2}{1-x} \\ \frac{dx}{1-x}=-\frac{du}{2u} \\...
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20 views

MCQ on Cauchy problem

$y u_x-xu_y=0,u=g $ on $ \Omega $ has a unique solution in neighborhood of $\Omega$ for every differentiable function g: $\Omega \rightarrow R$ if 1.$\Omega =\{(x,0):x>0\}$ 2.$\Omega =\{(x,y):x^...
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1answer
19 views

What does it mean when a system is made dimensionless and what is the exact technique for that?

For school research I'm working on a system of ODEs to describe a chemical oscillator (the Oregonator). This system is described with the following system: $$ \frac {dX}{dt}=k_1AY-k_2XY+k_3AX-2k_4X^...
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4 views

find the value of instablity error from which this value shows instability

I have an Euler method that has this form: $$\hat{I}(t_{n+1}) = \hat{I}(t_{n})+h\beta \hat{I}(t_{n})(1-\frac{\hat {I}(t_{n})}{N})$$ which can also be written like $$\hat{I}(t_{n+1})=\phi (\hat{I}(t_{...
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0answers
16 views

Is it always possible to get the direct function for a system of ODEs?

Is is possible for all ODEs to derive the direct formula? I'm wondering if there is some (very difficult) mathematical method to get the direct formula. As an example, consider the following system: ...
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1answer
26 views

The integral $\int \frac{1}{\sqrt[y]{y}} dy$ and the differential equation $y = \left(\frac{dy}{dx}\right)^y$

I couldn't find a question about this integral, sorry if a similar question has been asked before. For fun, I came up with the differential equation: $$y = \frac{dy}{dx}^{{\frac{dy}{dx}}^{{\frac{dy}...
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2answers
23 views

Is $dX/dt=X(t)$ the correct ODE for $X(t)=e^t$?

For a school project for chemistry I use systems of ODEs to calculate the concentrations of specific chemicals over time. Now I am wondering if $$ \frac{dX}{dt} =X(t) $$ the same is as $$ X(t)=e^...
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0answers
16 views

How to come up with Dulác-Function?

I'm currently studying dynamical systems and came across the Bendixson-Dulac-Theorem Let $D \subseteq \mathbb{R}^2$ open, $f \in \mathcal{C}^1(D, \mathbb{R}^2)$ and consider the nonlinear system $...
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1answer
26 views

Ordinary differential Equation , [closed]

The initial value problem $(x^2−x)\frac{dy}{dx}=(2x−1)y$, $y(x_0)=y_0$ has no solution if $(x_0,y_0)$ equals : Select one: A. $(1,1); \quad$ B. $(0,0); \quad$ C. $(2,1);\quad$ D. $(3,1)$.
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17 views

List of Laplace transform differential equations

I’ve been starting to learn about solving differential equations using the Laplace transform, and I wanted good, non obvious examples. I think this can help beginners(like me) get where it is ...

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