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 variable. For questions specifically concerning partial differential equations, use the [tag:pde] instead.
43,441
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How to alter my differential equations to force an abrupt decrease in the solution?
I have the following data sets:
...
- 1
1
vote
2
answers
42
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How to obtain $y(1)$ here?
Let $y$ be the solution if $y'+y=|x|,x\in \mathbb{R}$ , and $y(-1)=0$ then $y(1)$ equals
(a) $\frac{2}{e}$
(b)$\frac{2}{e^2}$
(c)$2e$
(d) $2e^2 $
Actually here $|x|$ is used in the question so I am ...
- 17.1k
1
vote
2
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74
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Brezis' exercise 8.23.4: how to prove $\|u\|_{L^p} \leq \frac{1}{k+\delta / (p p')}\|f\|_{L^p}$ in case $p \in (1, 2)$?
Let $I$ be the open interval $(0, 1)$ and $k >0$. I'm trying to solve a problem in Brezis' Functional Analysis, i.e.,
Exercise 8.23
Given $f \in L^1(I)$, prove that there exists a unique $u \in ...
- 16.7k
2
votes
2
answers
159
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Phase portrait of $\ddot{x} + \sin(x)=0$ near the origin.
I am trying to sketch the phase portrait of the second order ODE describing a pendulum near the origin:
$$\ddot{x}+\sin(x)=0.$$
I write this as the first order system:
$$ \begin{aligned} \dot{x} &=...
- 101
2
votes
1
answer
29
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There exists a unique stable limit cycle around $(0,0)$
Consider the system
$$
x'=\mu x-y-x\sqrt{x^2+y^2}, \, y'=x+\mu y-y\sqrt{x^2+y^2}
$$
I am working on that show that (1) as $\mu>0$, there exists a unique stable limit cycle around $(0,0)$ (2) as $\...
- 101
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47
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+50
How to show that is unique asymptotic stable
Based on this question:Poincaré-Bendixon show periodic solutions.
Show that the system $x^{'}=x-y-x^{3}$,$y^{'}=x+y-y^{3}$ has a unique periodic orbit on annulus $A:=\{(x,y): 1\le x^2+y^2\le 2\}$...
- 75
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29
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General approach on second order ODE
Consider a simple case in 1d such that
$$
-u''(x)+a(x)u(x)=0
$$
with Dirichlet boundary condition $u(x)=g(x)$. Both $a(x)$ and $g(x)$ are $C^2$ functions. Is there any theorem on the analytic solution ...
- 125
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17
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Why the 0-order coefficient of Legendre equation should be nonnegative
When we try to find the solutions of Legendre equation
$$ (1-x^2)y''-2xy'+\lambda y=0 , \quad y|_{x=\pm 1}<+\infty$$
we may let the 0-order coefficient $\lambda\geq 0$, this conclusion could be ...
- 21
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Application of the Poincaré-Bendixson theorem
I am trying to solve the following exercise (19) from this magnificent notes but I am encountering some problems:
Prove that the system
$$\begin{cases}
\dot{x} = 2x-x^5-xy^4 \\
\dot{y} = y-y^3-yx^2
\...
- 71
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0
answers
28
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Doubt on the solution I got from 2D harmonic equation
$f$ is function that $f(0)=1$ and $f:[0,\infty)\to\mathbb{R}$. It's continuous on $[0,\infty)$ and is $C^2$ on $(0,\infty)$. Let $u$ be another function defined on $\mathbb{R}^2$ and $u(x,y)=f(\sqrt{x^...
- 676
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18
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Coupled Differential Equation Involving Trig Functions
Given the initial conditions $r=L$ at $\theta=0$, solve $r(t)$ and $\theta(t)$ for the following set of coupled differential equations.
$$
\dot{r} = -u - v \sin \theta
$$
$$
r \dot{\theta} = -...
- 829
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1
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19
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How to prove the given statement based on the conditions related to interpolating polynomials used in Adaptive Backward Differentiation Formula.
Full disclosure: This was part of my homework that I couldn't solve completely. Also, I wasn't sure how to give a proper title to this question.
Before going to the statement to prove, I think some ...
- 124k
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1
answer
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Showing that every plane curve with constant curvature is a circle by solving the Frenet differential equation
I recently saw the following example that shows that plane curves with constant curvature are circles:
Let $\kappa \neq 0$ be constant and $s \in \mathbb{R}$ arbitrary, then by the Frenet Matrix-...
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24
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0
votes
2
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32
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Substitution on a Temperature Problem
Given real parameters $A,B,C$ consider the temperature problem with non-homogeneous boundary conditions:
$$u_t=ku_{xx}, \;\;\; u=u(x,t), \;\;\;0\leq x\leq \pi,\;\;\; t,k> 0$$
$$u_x(0,t)=u(0,t)+...
- 85.9k
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36
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Need help finding why there no equilibrium solutions to this function
Why does the function $\Bbb dy/\Bbb dx=\sin(x)\cos(y)+\cos(x)$ have no equilibrium solutions? I have already thought of the fact that for $\sin(x)$ to equal 0, $\cos(x)$ would equal 1 and that rules ...
- 3,874
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1
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Is $f'(x)=f(1/x)$ solvable?
So recently I have been scrolling through Youtube (mainly to find math videos for entertainment, I'll attempt a question on my own every now and then) when I came across this video by Michael Penn ...
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23
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6
answers
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Can someone intuitively explain what the convolution integral is?
I'm having a hard time understanding how the convolution integral works (for Laplace transforms of two functions multiplied together) and was hoping someone could clear the topic up or link to sources ...
- 131
3
votes
2
answers
489
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Stability of equilibrium of a nonlinear system of ODE's
Suppose we have the nonlinear system of ODE's
$$\begin{cases}
\dot{x_1} = -\beta x_1 x_2 \\
\dot{x_2} = \beta x_1 x_2 - \gamma x_2
\end{cases}
$$
Where we take $\beta, \gamma > 0$ arbitrary for ...
- 101
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vote
3
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How is this differential equation simplified? Is $\frac{d(v_a + v_b)}{dt} = \frac{dv_a}{dt} + \frac{dv_b}{dt}$ with $v_a$ and $v_b$ functions of $t$?
How can this equation
$$
(v_i - v_1)g_1 - C_1\frac{dv_1}{dt}+C_2\frac{d(v_o - v_1)}{dt} = 0
$$
be simplified to this?
$$
v_i g_1 = g_1 v_1 + (C_1 + C_2)\frac{dv_1}{dt} - C_2\frac{dv_o}{dt}
$$
I see ...
- 3,324
1
vote
1
answer
32
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General method for finding invariant subsapces of a nonlinear system
Suppose we are given a system:
$$\dot{x_{1}} = f_{1}(x_{1},...,x_{n})$$
$$...$$
$$\dot{x_{n}} = f_{n}(x_{1},...,x_{n})$$
And are interested in finding subspaces of the vector space that are invariant ...
- 4,087
-1
votes
1
answer
71
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Solve the Partial Defferential Equation $z+xp-x^2 y q^2-x^3 p q=0$, where $p=\frac{\partial z}{\partial x}$ and $q=\frac{\partial z}{\partial y}$.
I use Charpit Method to solve the problem but calculation is so big. Is there any another method to solve the problem.I think I need some variable transformation which gives me a standard form then I ...
- 12.1k
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14
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Eigenvalue problem of an Ordinary Second order Linear Homogeneous differential equation [closed]
Do we have a closed form solution for a canonical form OSLH equation?
$$
y^{\prime \prime}+(C+x\sin(x))y=0
$$
where C is a constant or the eigenvalue. I have checked Andrei D.Polyanin's book "...
- 27
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votes
1
answer
123
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Proof of Peano existence theorem in ZF without mathematical logic
There is a proof of Peano existence theorem in ZF.
Peano existence theorem:
For any open $D \subseteq \mathbb{R}^2$, continuous $f:D \to \mathbb{R}$ and initial condition $\langle t_0,x_0\rangle \in ...
- 40k
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2
answers
74
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Can you help with this exponential decay question?
Suppose that 100 kg of a radioactive substance decays to 80 kg in 20 years.
a) Find the half-life of the substance (round to the nearest year).
b) Write down a function $y(t)$ ($t$ in years) modeling ...
- 3,874
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vote
1
answer
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$y''=a\frac{y}{(y')^2+b},y(0)=0,y'(0)=c,\forall c\in\mathbb{R}$
Can some solve the following equation or at least tell me how to approximate the result?
$$y''=a\frac{y}{(y')^2+b},y(0)=0,y'(0)=c,\forall c\in\mathbb{R}$$
I have no idea if it is an easy task, but for ...
- 56.6k
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0
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17
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Difusion - line methods - dissolution of minerals
Could someone assist me to solve this problem using the lines method? I have a 1D model that describes solute diffusion across a system with two layers: a fluid boundary layer and a porous secondary ...
- 3,874
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0
answers
36
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Bijectivity and smoothness of nonautonomous ODE solution
Let $f: \mathbb{R}^n \times [0, T] \rightarrow \mathbb{R}^n$, and define the ODE:
$d x(0) = f(x(t), t) dt$ with initial condition $x(0) = x_0$.
Assuming a unique solution, the solution characterizes a ...
- 381
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Checking the Existence of solution to the DE given initial values
The ODE is as given below:
$$\frac{dy}{dx}=\frac{yx^2-xy^2}{x^3};y(0)=0$$
Following the rule, we take $f(x)=\frac{yx^2-xy^2}{x^3}$
Checking the continuity of $f(x)$, we have;
$$\lim_{(x,y)\to(0,0)}{\...
1
vote
1
answer
42
views
Transform Lagrangian with a square root
I have an action given by,
\begin{equation}
S = \int^{\tau_f}_{\tau_0} d\tau \left(\frac{1}{z^3}\sqrt{-f(u,z) \dot{u}^2 + 2 \dot{u} \dot{z} + \dot{x}^2} + \frac{2 \dot{z}}{z^3} \right),
\end{equation}
...
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0
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27
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Describe bifurcation for $\mu \in \mathbb{R}$ and draw the diagram
I am working on the question that describe bifurcation for $\mu \in \mathbb{R}$ and draw the diagram.
$$
%
\begin{align}
%
\dot{x} &= -x^4 +5\mu x^2+4\mu^2:=f(x,\mu) \\
%
\dot{y} &= -y \\
%
\...
- 75
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votes
1
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29
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Use the Fundamental Theorem of Calculus to find a solution to the initial value problem
Let's consider the initial value problem: $ dy/dx = x - 2, y =(0) = 1 $.
Part 1 is to find the equation of the tangent line at x = 0 which is $y = -2x + 1$.
Part 2 I am given 3 more tangent lines:
(1, ...
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Prove that the solution to a cauchy problem is monotonically decreasing
Considering the cauchy problem
\begin{cases}
x' = 5\cos^2 (tx)-x^2 -5\\x(t_0)=x_0
\end{cases}
How can i prove that if $x_0 \neq 0$ then the solution $x(t)$ is monotonically decreasing and then find ...
- 1
1
vote
1
answer
31
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How to find a particular solution of a differential equation with a fractional function of x?
I am finding $y$ that satisfies
$$y^{(4)}-81y = \frac{12}{x^5}-\frac{40.5}{x}.$$
First, we can easily find $y_h$ as follows:
The characteristic function can be derived as $\lambda^4 - 81$=0.
From the ...
- 3,355
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0
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25
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Find the definite solution [closed]
Find the definite solution of the following.
Given
$$
\begin{cases}
C_t= 300+0.5Y_t + 0.4Y_{t-1}\\
Y_t= 200+0.2Y_{t-1} \\
Y_o= 65~000
\end{cases}
$$
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1
answer
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Inhomogeneous heat equation with Fourier transform
Consider the heat equation
$$\dfrac{\partial u}{\partial t}=\dfrac{\partial^2 u}{\partial x^2} +G(x,t).$$
with the condition $u(x,0)=f(x)$. When $G(x,t)=0$ it is quite easy to solve it using Fourier ...
- 1,266
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0
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15
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Legendre Differential equation, n(n-1) or n(n+1)
I am confused regarding the Legendre Differential Equations' coefficients.
In some books its,
$(1-x^2)y''-2xy'+n(n-1)y=0$
and somewhere it is,
$(1-x^2)y''-2xy'+n(n+1)y=0$
what is its correct form?
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+50
Closed form expression for an integral with a cosine function
I am attempting to solve a first-order ODE of the form:
$$y'(x) = {\rm sech}^2(x){\rm sech}'(x) \times(a + b\cos(cx))$$
The part with the constant was easy to solve and I get
$$y(x) = a\int {\rm sech}^...
- 256k
-1
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0
answers
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views
Find a function that does not satisfy that IVP [closed]
Find $f\in C^{\infty}(\Bbb R,\Bbb R)$ s.t IVP has not satisfied $f(x')=0$ and $x(0)=0$.
Try:
I found a paper that talks about that and gives an example but from $\Bbb R^2$ to $\Bbb R$, on page 347, I ...
- 531
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4
answers
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Conversion of rotation matrix to quaternion
We use unit length Quaternion to represent rotations. Following is a general rotation matrix obtained ${\begin{bmatrix}m_{00} & m_{01}&m_{02} \\ m_{10} & m_{11}&m_{12}\\ m_{20} & ...
2
votes
1
answer
48
views
Find the solution of $2y(x−1)y′ +y^2 = 4x(3x−2)$ with initial condition $y(1)=2$ and hence find the maximal interval interval of the solution.
Find the solution of $2y(x−1)y′ +y^2 = 4x(3x−2)$ with initial condition $y(1)=2$ and hence find the maximal interval interval of the solution.
My attempt:-
We get the solution of the differential ...
- 2,029
1
vote
0
answers
31
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Flow of dynamical system evolves towards its interior guaranteed by imposing strict inequality in Nagumo's theorem?
To prove a closed set is positive invariant (i.e. the flow of ode either tangent to the boundary of the set or point inwards the interior of the set), we could use Nagumo's theorem. Consider the ...
- 519
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votes
1
answer
53
views
Help Solving and Understanding a Temperature Problem
Consider the following temperature problem:
$$u_t(x,t)=ku_{xx}(x,t), \;0\leq x \leq \pi,\;\; t,k >0$$
with boundary conditions:
$$u_x(0,t)=u(0,t)$$
$$u_x(\pi,t)=u(\pi,t)$$
$$u(x,0)=f(x)$$
I know ...
- 85.9k
0
votes
0
answers
26
views
Solving a second-order one-dimensional ordinary differential equation numerically using FEM with nonlinear meshing [closed]
can I solve a second-order one-dimensional ordinary differential equation numerically using FEM with nonlinear meshing or non-linear discretization? Or the meshing must be linear?
$$
y''+y=f(x),
$$
...
- 21
2
votes
0
answers
31
views
Non-linear spring modelling
My question probably sits more on the applied mathematics side.
There is a spring mass system, for the case where the mass moves into the spring (vertically downwards) the spring experiences ...
- 21
0
votes
0
answers
36
views
Modelling Peters Swing from the Becoming Spider-Man Scene - The Amazing Spider-Man (2012)
I am currently working on a project where I model the physics and calculus involved in a successful swing done by spiderman. I have broken down his swing into three segments. The first segment models ...
- 11
1
vote
1
answer
51
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Runge Kutta 4th order with coupled equations where derivatives are linked [closed]
I have 2 coupled second order equations as below:
$$ \ddot{y}(t) + a\dot{y}(t) + by(t)=q(t) $$
$$ \ddot{q}(t) + c\dot{q}(t) + dq(t) = A\ddot{y}(t)$$
I'm wondering if it is possible to solve this ...
- 11
0
votes
1
answer
53
views
Sylvester-like equation solution
I would like to solve a coupled matrix differential equation.
All are $2\times 2$ matrices. Then, I have
\begin{align}
&\dot{X}=-i (A X - X A)-\eta (B Y - Y B);\\
&\dot{Y}=-\kappa Y +(B ...
- 34.4k
0
votes
0
answers
36
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How far does the connection between the boundary and the derivative go?
The generalized Stokes theorem $$\int_M d\omega=\int_{\partial M} \omega$$ for a compactly supported differential form $\omega$ on a compact manifold with boundry $M$ establishes a sort of duality ...
- 1,401
3
votes
1
answer
45
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Confusion regarding the definition of the state of a physical system
I'm currently covering Jan de Vries' Elements of Topological Dynamics and in it he gives a brief introduction to the field through the lens of classical mechanics. He defines the state of a mechanical ...
- 10.5k