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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Finding the lower bound for Radius of Convergence of a Frobenius Series

this is my first post so I'll try to be brief. I am working on finding a problem such as this: given the ODE: $$(x^2-1)y''+x^2y' + cotx\cdot y = 0$$ "Find all the singular points of the following ...
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3 answers
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A difference equation or functional equation that has a solution $(ax+b)^c$

I am looking for a (system of) functional equation or difference equation that has a solution: $f(x)=(ax+b)^c$, where $a,b,c$ are constants. Constant $a,b,c$ cannot appear in the equation. For example,...
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Find the difference equation given the general solution $y(k) = c_{1}5^{k} + c_{2}(-5)^{k} + c_{3}6^{k}$

Given that $y(k) = c_{1}5^{k} + c_{2}(-5)^{k} + c_{3}6^{k}$, is the general solution to a difference equation, how do you work backwards to find the difference equation?
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Decide whether given function is the general solution to ODE

I have a set of function and must decide whether each of them is a general solution to an ODE of the form : \begin{cases}y'(x)=f(x,y(x))\\y(x_0)=y_0\end{cases} When the an ODE comes obvious for a ...
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Why is the fundamental matrix not singular for all t

Book: Linear System Theory and Design Author: Chi-Tsong Chen Page: 107 counter example $X(t) = \left[{\begin{array}{cc}1 & 1\\1 & e^t\end{array}}\right]$ Let $t_1 = 0$ and $v = \left[{\begin{...
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exist positive solution to the second order linear differential equation

Assume $\varphi :(a,b) \to \Bbb{R}$ which is smooth and satisfies some second order linear ordinary differential equation with smooth coefficient:$L\varphi = 0$, assume also that $\varphi(t_0) = 0$ ...
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help with understanding the proof for: "Fundamental matrices are non-singular for all t"

Some definitions Definition 4.1, An n$\times$n matrix $\Phi$ with the property that its columns are solutions of $\dot X = AX$, and lineraly indepenedt at $t_0$, is said to be a fundamental matrix of $...
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Equality of functionals/relations upon interchanging arguments implying equality of functions?

Say we have a pair differentiable functions $Z(z)$ and $R(r)$ of the real variables $r,z$. We also have another pair of functions $\hat{Z}(z)$ and $\hat{R}(r)$ of the same variables. We have an ...
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Is there any theorem for the solutions $u(t)$, $v(t)$ of the following differential equation? [closed]

Suppose I have the following equation (where $a(t)$, $b(t)$ and $c(t)$ are continuous functions) $a(t)(u'(t))^2 + b(t)u'(t)v'(t) + c(t)(v'(t))^2 = 0$ Is there any theorem that could possibly tell me ...
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Rewriting system of second order differential equation as system of first order

Given a charged particle moving in an electromagnetic field. We have $N$ amount of point charges placed in $\mathbb{R}^2$ on the coordinates $p_i$. We also have a free particle moving in $\mathbb{R}^2$...
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$x'(t)=|t-1|x(t)+1,x(0)=0$ Find $x(2)$ (using integrals is possible)

$x'(t)=|t-1|x(t)+1,x(0)=0$ Find $x(2)$ (using integrals is possible). First,how am I suppose to use $x(0)=0$ if I find the solution for $t>1$ and $t<1$ and I have to find $x(2)$ ? My solution : ...
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Singular perturbation differential equation

Find solutions for this system of differential equations with singular perturbation $$x'(t)=y, x(0)=x_0 \\ \epsilon y'(t)=\pm y, y(0)=y_0$$ with $t\geq 0, 0< \epsilon \ll 1$ for both cases ($\pm$). ...
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Quadratic diferential equation [closed]

Given constants $C$, $v$ and $k$, consider the following ODE $$\dot u = C \left( v^2 - 2 u v + u^2 \right) - k$$ I have to find $u$. Because the equation is a first order non-linear differential ...
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1 answer
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How many functions are possible for the given differential equation?

There was this question asked in a competitive examination , the solution of which is very confusing to me. The number of differentiable functions $y:(-\infty, \infty) \rightarrow [0, \infty)$ ...
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Differential equation (encyme reaction)

Consider this enzyme reaction with initial conditions $c_1(0) = c_2(0) = p(0) = 0,s(0) = s_0, e(0) = e_0, e^{\ast}(0) = e^{\ast}_0.$ I determined this differential equations for the enzyme reactions: ...
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how do i solve non linear equation? TOPIC IS PDE [closed]

Find a separated solution of the following nonlinear wave equation: ∂u/∂t=cu ∂y/∂x and What is a separated solution of the 2 -dimensional wave equation (∂^2 u)/(∂t^2 )=a (∂^2 u)/(∂x^2 )+b (∂^2 u)/(∂y^...
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1 answer
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An inequality for a maximal solution of an IVP [closed]

We have the function $f : \mathbb{R} \times \mathbb{R} \to \mathbb{R}, (x,y) \mapsto \frac{xy}{\sqrt{y^2+1} }$ and the following IVP \begin{align*} y'=f(x,y), \qquad y(0)=1. \end{align*} How does one ...
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Solution of 2nd order ODE [closed]

How can I solve the following ordinary differential equation (ODE) analytically? $$ a(u)\frac{d^2u}{dx^2}+\left(\frac{du}{dx}\right)^2 \frac{da(u)}{du}+b =0$$
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Realizing the Hopf fibration as the phase space of a small oscillations spherical pendulum

Page 35's of Arnold's ODE book "The deviation from the vertical is characterized by two numbers $x$ and $y$. It is known from mechanics that the equations of small oscillations have the form: $\...
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Any orbit of $X$ passing through a point of $M$ is entirely contained in a surface

I am trying to prove this result: Given a field $X$ in $\mathbb R^n$ of class $C^1$ and let us consider a surface $M\subset\mathbb R^n$ of class $C^2$ such that $p\in M$ implies $X(p)\in T_pM$. Show ...
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Gradient of an objective function containg coupled odes

I have position values of the form $\mathbf{X} = \left[x \; \;y \right]^{T}$. Frenet-Serret model for 2D would consist of following equations: $$ \frac{d{\mathbf{X}}_{model}}{dt} = V(t){\mathbf{T}} $$ ...
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Solve the following differential equation $t + \frac{f(t) f'(t) \sqrt{n^{2}+n^{2}f'(t)^{2}-1}-f(t)}{\sqrt{n^{2}+n^{2}f'(t)^{2}-1}-f'(t)} = const.$

I have to find a real function $f(t)$ which makes the following $s$ constant : $$s = t + \frac{f(t) f'(t) \sqrt{n^{2}+n^{2}f'(t)^{2}-1}-f(t)}{\sqrt{n^{2}+n^{2}f'(t)^{2}-1}-f'(t)}$$ where $n$ is a ...
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Showing inequality for the norm of an integral equation

Let $\xi \in \mathbb{R}^2$, $\Phi \in C^0([0, +\infty[, \mathbb{R}^{2x2})$ a bounded function. Let $y:[0, +\infty[ \rightarrow \mathbb{R}^2$ a solution of the following integral equation, $$ y(x) = e^{...
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Sturm-Liouville problem: why $\lambda < 0$ and how to fix it?

I'm trying to solve the following problem (with $a \in \mathbb{R}^+$ and $\lambda \in \mathbb{C}$): \begin{equation} \begin{cases} a^2y'' - \lambda y = 0\\ y(0) = y(\pi) = 0 \end{cases} \...
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What is the solution to $y'=p(x)+y $ where $p(x) =\sum_{k=0}^d p_kx^k $?

This is a generalization of Power series solution of differential equation $y'=x^2 +y$ Show that the solution to $y'=p(x)+y$ where $p(x) =\sum_{k=0}^d p_k x^k$ is $$ y(x) =ce^x- \sum_{j=0}^d \...
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1 vote
0 answers
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Why the poincaré theorem is not possible in $\mathbb R^n$?

Hello I am reading about the Poincaré-Bendixson theorem in the plane and then in compact two-dimensional manifolds. But I have some doubts that I would like you to help me clarify: I know the example ...
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1 answer
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Power series solution of differential equation $y'=x^2 +y$

so I'm working to find a power series solution for diffrential equation : $y'(x)=x^2 + y(x)$ with given $y(0)=1$ and and i have to prove that the power serie is a solution to this equation. i did ...
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-7 votes
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Draw a full phase portrait for the following non-linear system [closed]

You need to draw a portrait phase of non linear systems $x′=−y−x^2+1$ $y′=−x+y^2−4$
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2 votes
2 answers
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Solution (recurrence relation) of non-linear DE using the method of power series

I have to solve this non-linear DE $y' -e^y -x^2 = 0 , y(0)=c$ using powerseries. $y(x) = \sum_{n=0}^\infty a_{n}x^n $ $y'(x) = \sum_{n=1}^\infty na_{n}x^{n-1} $ so we get $\sum_{n=1}^\infty na_{n}x^{...
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O.D.E. with elliptic integral

I have an energy integral $W=\int V dx$, where after using spherical parametrization $V$ is $V=\frac{1}{2}[(\theta'-\lambda)^2+\gamma^2\cos^2\theta]$. Stationary points of the equation, using Euler -...
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Euler integration solution from system of ODE's - already estimated values

I am currently completing an investigation assignment on modelling the growth of a virus inside of the host. There are 3 ODEs that I am using in the system, all determined by change in t. The ...
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A basic question on the introduction of contracting system

I am learning contraction theory from this tutorial which starts by calculating the difference between two arbitrary trajectories of a scalar system (corresponds to 26:30 in the video): the ...
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2 votes
1 answer
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fourth-order finite difference for $(a(x)u'(x))'$

Previously I asked here about constructing a symmetric matrix for doing finite difference for $(a(x)u'(x))'$ where the (diffusion) coefficient $a(x)$ is spatially varying. The answer provided there ...
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Input for stopping in Dubin's path dynamics?

This question may seem pretty dumb. But I really want to know. I have the following linear dynamical system for Dubin's path, \begin{align*} \phi &= \begin{bmatrix} ...
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Transform initial condition between two systems of first order ODEs

I encountered a problem when preparing for my upcoming exams. I have a system of first order ODEs $$\dot{X}=AX+B(t)$$ where $A=\begin{pmatrix}-2 & 4\\-4 & -2\end{pmatrix}$, $B(t)=\begin{...
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1 answer
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Determining a function which fits the given differential equation

Is it possible to find a function $f(x)$, which satisfies: $$\frac{d}{dx} f(x) = \frac{\int f(x) \: dx}{x \cdot f(x)}$$ If so, how?
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How can we derive $u_0 (t) = a_0 \cos (\omega_0 t + a_0)$ from this solution?

I am reading the Mason, D. P., On the method of strained parameters and the method of averaging, Q. Appl. Math. 42, 77-85 (1984). ZBL0545.70033 and on page $79$ solves the very trivial differential ...
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How to prove Boundedness of two 3D coupled chaotic c=system resulting into 6D system? [closed]

I have a two 3D chaotic system and I couple them to make a 6D system. How could I prove boundedness of the coupled nonlinear differential equation? The system description is like this: xdot(t)=f1(x(t))...
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4 votes
2 answers
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An interesting recurrent equality, possibly easier to solve in its differential form?

I encountered an interesting inequality that I'm not sure how to approach. Here $c$ is a positive constant. $$f(n+1) - f(n) = c f(n)\sum_{m=0}^n f(m)$$ I am not familiar with techniques to solve ...
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3 votes
1 answer
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"Spanning" of solutions of ordinary differential equations

Suppose we have a switched ODE $$\dot{x} = A_{\sigma(t)}x,$$ where $A_{\sigma(t)}$ is a constant matrix given $\sigma(t)\in\mathcal{M}=\{1,2,\cdots,m\}$. If we fix the initial condition and can ...
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Show that the drainage is defined by this ode

A water tank with a rectangular cross-section and one trapezoidal side as shown in the figure contains a volume of water $𝑉(ℎ)$ [$\text{m}^3$] when the depth of the water in the tank is $ℎ ~\text{m}$ ...
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Second order nonhomogeneous differential equation

Let's suppose to have a system of four first order nonhomogeneous differential equations that i can regroup into a sysyem of two second order nonhomogeneous differential equations. Then I can also ...
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0 answers
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Identify if a given expression is a general solution of an ODE

Let the following expressions: $$ \text{(i)}: \lambda(x; x_0,y_0) = y_0 \cosh(x - x_0) \\ \text{(ii)}: \lambda(x; x_0,y_0) = y_0 \tan^2(x - x_0) \\ \text{(iii)}: \lambda(x; x_0,y_0) = (3[x^2 - x_0^2] +...
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To determine the type of a singularity of a 2-D dynamic system.

This is a problem in my course homework. Suppose that $a>0$, $b>0$, considering the following 2-dimensional autunomous system: $$\begin{align}\frac{dx_1}{dt}&=F(x_1)=a-x_1-\frac{4x_1x_2}{1+...
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66 views

Solve $x \frac{d^{2}y}{d x^{2}}+\frac{d y}{d x}+x y=0$

Solve the differential equation $$x \frac{d^{2}y}{d x^{2}}+\frac{d y}{d x}+x y=0$$ My try: Let $z := x \frac{dy}{dx}$. So, we get $$\frac{dz}{dx}=x\frac{d^2y}{dx^2}+\frac{dy}{dx}$$ and, thus, $$\frac{...
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-1 votes
0 answers
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Can we solve this differential equation or there is some other way [closed]

$f(x)$ is defined for $x≥0$ and has a continuous derivative. It satisfies $f(0)=1,f'(0)=0$ and $$(1+f(x))f''(x)=1+x$$ Then which of the following is not a possible value of $f(1)$? $2$ $1.75$ $1.5$ $1....
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2 votes
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Does the solution to the following differential equation exists?

I am a physics student and I encounter a the following differential equation for the function $\phi(r)$ (scalar field on Kerr metric) $$\frac{-2\phi-2(M-r)\phi'+(P^2-2Mr+r^2)\phi''}{r^2}=-m^2\phi$$ ...
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1 vote
1 answer
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How to show that the solution of a delayed differential equation is unique? Can you use Picard Lindelof theorem?

So I know that you can use Picard Lindelof to show uniqueness of a solution to an ODE, but if given an ordinary delayed differential equation, could you still use it? How would that establish ...
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1 vote
1 answer
41 views

Numerical Differentiation Table

The following data was collected by measuring the distances in kilometres that a moving object travels over time (t) in seconds t 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 s 0.0 9.0 20.0 34.0 48.0 64.0 80....
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1 vote
1 answer
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linearization of non-linear ODE

We have a non-linear ODE of the form $ \dot x= f(x, h(x))$ where $g(x, h(x))=0$, $h: X\subseteq\mathbb R^n\to \mathbb R^m$, $g:\mathbb R^n\times \mathbb R^m\to \mathbb R^m$, $f:\mathbb R^n\times \...
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