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

learn more… | top users | synonyms (1)

2
votes
0answers
55 views

Estimating limit cycle of ODE system

I'm looking at a system of ODEs: $$\dot{x} = -y - \epsilon^2 x + xy^2$$ $$\dot{y} = x -\epsilon^2 y - x^2$$ After plotting these in Matlab I can see there is a limit cycle very close to the origin ...
2
votes
0answers
58 views

Solving first order non linear ODE

I am trying to solve the following first order non-linear differential equation: $$ \frac{\partial y}{\partial x} = -\sqrt{\frac{2(\sigma + 1)}{\sigma}} \sqrt{-\frac{1}{2y^{2}} + ...
2
votes
0answers
50 views

How to solve this non-linear, second order ODE

does anyone know how to solve this ODE? $ yy'' +y' +y =0 $ where y is a function of one real variable.
2
votes
0answers
32 views

sufficient conditions for finite time of existence of integral curves of a vector field

Let $U\subset \mathbb{R}^2$ open, $\partial U\neq \varnothing$, $V\colon U\rightarrow \mathbb{R}^2$ smooth. Let $c\colon [0,t_{max})\rightarrow U$ be an integral curve of $V$, where $t_{max}$ is the ...
2
votes
0answers
77 views

How to solve this initial-boundary value problem for a PDE

Consider $$u_{tt}-a^2u_{xx}+u_t+a u_x=0,\quad 0<x<\infty,\quad t>0,(*)$$ where $u_t=\frac{\partial u}{\partial t}$ and etc. It is not so hard to use the method of characteristics to solve it ...
2
votes
0answers
58 views

How to solve the ODE $2x\frac{dy}{dx}=C(1+(\frac{dy}{dx})^2)^2$?

I am struggling with this ODE I obtained when solving the Euler-Lagrange equation. Can any one help me with solving the ODE $$2x\frac{dy}{dx}=C(1+(\frac{dy}{dx})^2)^2$$ Thanks so much! It comes ...
2
votes
0answers
42 views

Time taken to empty a hemispherical shaped tank

The tank has a radius of $2$m when initially filled and has an outlet of cross section $12$ cm2 Outlet flow as I calculated goes according to the law $V(t)=0.6\sqrt{2gh(t)}$. Having found out the ...
2
votes
0answers
211 views

How to solve this recurrence relation and solving the power series

Take the following recurrence relation into account: $$ a_{n+2} = \frac{1}{(n+1)(n+2)} \sum_{k=0}^n (s_k - (k+1)a_{k+1})(n-k+1)a_{n-k+1} $$ I know that: $$ s_{2m+1} = \frac{(-1)^m}{(2m+1)!} $$ and ...
2
votes
0answers
43 views

How to solve $\int_{x}^{x+a} f_X(u) du=e^{-2\lambda_1 x} \int_{x-a}^{x} f_X(u) du$

How to solve equation of the type \begin{align*} \int_{x}^{x+a} f(u) du=e^{-\lambda x} \int_{x-a}^{x} f(u) du \end{align*} we want to solve for $f(x)$ where $\lambda,a$ are some constants. Things I ...
2
votes
0answers
23 views
2
votes
0answers
34 views

How to find first-order quasi-linear PDEs form second-order quasi-linear PDE?

Transform $u_{tt} u_{xx}-u^{2}_{tx} + uu_{tt} + 1=0 $ into first-order quasi-linear PDEs. Attempt: $u_{tt}(u_{xx}+u)=(u_{tx}-1)(u_{tx}+1)$ To get $u_{tt} = u_{tx}-1\Rightarrow u_t = u_x ...
2
votes
0answers
352 views

Differential Equation Direction field

What i want to achieve: I want to plot the direction fields of the following three differential equations: 1. Malthusian growth model: $p'(t)=\lambda*p(t)$ with $\lambda=1$ and $p(t)=t$ 2. Linear ...
2
votes
0answers
19 views

Convergance of DASPK for a non-linear DAE

I have a system of non-linear DAE and I noticed that the system does not converge if some of the equations are not differentiated. For example, if the control volume equation is represented as this: ...
2
votes
0answers
34 views

What is the general way to rewrite an ODE with respect to a change in coordinates?

I have an ODE : $y' = f(x, y)$ I change for coordinates $(r, s) = (g(x, y), h(x, y))$. What is the equation like in terms of r and s ? If it can help, in my case, $(r, s) = (y.x^{-k}, ln(x))$. I can ...
2
votes
0answers
8 views

Differential Equations of Transformed System

At the moment I'm struggling with a problem I found in a script to one of my lectures: Let $\phi \in C^\infty(\mathbb{R}^{2n})$ have the property that the system $p_i=\frac{\partial}{\partial ...
2
votes
0answers
64 views

A line integral equation popped up when trying to derive Exact ODE integrating factor, can it be solved analytically?

(For convenience, for any functions, only its first instance the x,y dependence will be written out, all subsequent instance the x,y will be suppressed) I have an ODE $$M(x,y)+N(x,y)\frac{dy}{dx}=0$$ ...
2
votes
0answers
56 views

book suggestion on manifolds

I've to learn differential equations on Manifolds. Can any one please suggest some books/lecture notes for differential equations on Manifolds ?
2
votes
0answers
45 views

Stability of non-autonomous stochastic differential equation

I'm looking for a good reference or insight to under what conditions can I prove stability (or instability) for the following general n-dimensional non-autonomous stochastic differential equation: ...
2
votes
0answers
32 views

Determining linear independence of three simple functions for a third order ODE. (2.9-7)

This is a very similar post to one previous by me but I felt that not all questions were satisfactorily answered. But I am sincerely grateful to those who tried. I would like a sharp independent eye ...
2
votes
0answers
51 views

suggestion for lyapunov function

Consider differential equation \begin{align}x'&=-t(x+y)\\ y'&=-y+x-y(y^2-6).\end{align} Can some one suggest a lyapunov function for it. I have examined $V(x,y)=x^2+y^2$ , ...
2
votes
0answers
141 views

Find the critical curves for the following functional

Find the critical curves for the following functional : $$J[y(x),z(x)]=\int_{0}^{1}(y'^2+z'^2-xyz'-yz)dx$$ With the conditions : $$K[y(x),z(x)]=\int_{0}^{1}(y'^2-xy'-z'^2)dx=2$$ $$y(1)=z(1)=1$$ ...
2
votes
0answers
41 views

Is this a valid equivalence between the classes of Differential Equations?

Consider the general first order Linear Ordinary Differential Equation: $$ \frac{dy}{dx} = A(x,y) = \frac{F(x,y)}{G(x,y)}$$ This equation is characteristic equation of the Partial Differential ...
2
votes
0answers
43 views

Two linear independent functions have zero Wronski

Suppose $f$ and $g$ are linear independent $C^1$ functions on $[a,b]$ and there Wronski det is zero, i.e. $$fg^{'}-f^{'}g=0$$ Can we say there exist an point $t_0\in [a,b]$ such that: ...
2
votes
0answers
23 views

Why is the basin of an asymptotically stable solution to a differential equation an open set?

Why is the basin of an asymptotically stable solution to a differential equation an open set? The basin is defined as the set off all points s.t. their limit at $t = \infty$ is equal to some solution ...
2
votes
0answers
73 views

Pursuit Curve, Parametric Equation

So its a classic problem: Object $A$ starts at the origin $(0,0)$ and moves straight up the $y$ axis with a speed $v$. Object $B$ starts at point $(1,0)$, always moves towards object $A$ and has a ...
2
votes
0answers
83 views

a system with differential equations

I have a system which is described by the following differential equation. I want a closed form formulas to calculate $v_1(t)$ and $v_2(t)$ with the given parameters. In the following equations $p, k, ...
2
votes
0answers
61 views

Recursive differential equations

Suppose we took the odd solution to $y''+y=0$ which is $\sin(x)$. If we put this in place of $y$ in the differential equation we get the equation: $$y''+\sin(y)=0$$ the odd solution to which is an ...
2
votes
0answers
150 views

best book for practicing unsolved problems in differential equation and linear algebra

I started reading differential equation and linear algebra. Can anyone provide the link/book name where I may get many questions to practice. Generally, in the end of book only few problems are there. ...
2
votes
0answers
62 views

Equilibrium points and linear stability

Consider the nondimensional amplitude equation for $A = A(t)$ where $t$ is time given by (1): $$ \frac{dA}{dt} = \sigma A - a_1 A^3 - a_3 A^5 = f(A) \text{ with } \sigma \in \mathbb{R}, a_1 < 0, ...
2
votes
0answers
51 views

Why does this nonlinear ODE solution not work?

I am relatively new to Python and trying to use it to solve a second order nonlinear differential equation, specifically the Poisson-Boltzmann equation in an electrolyte. $$\phi''(r) + \frac2 ...
2
votes
0answers
274 views

Identifying Hamiltonian Systems with Phase Portrait

the following is a homework question (that isn't going to be graded) and I'm not sure how to do it. I know that the solution trajectories cannot cross the H(x,y)=constant curves, but I'm not sure ...
2
votes
0answers
40 views

BVP eigenvalue problem

I am working on the following problem and I am completely stuck: Show that the eigenvalue problem $$ -u''+4\pi^{2}\int_{0}^{1} u(x)\,dx=\lambda\,u $$ with $u(0)=u(1)$ and $u'(0)=u'(1)$ has ...
2
votes
0answers
33 views

Stability of an equilibrium solution with 0 denominator

I'm testing the equilibria of a differential equation and found that one has a 0 denominator. Example: $$\frac{dx}{dt}=2x^{(1/2)}-5$$ Which, when you try and evaluate the derivative at 0, you end up ...
2
votes
0answers
132 views

Stuck trying to solve wave equation in $n$-dimensions.

Solving the wave equation $u_{tt} = c^{2} \Delta{u}$ subject to $u(0,x) = f(x)$ and $u_{t}(0,x) = g(x)$ gives us d'Alembert's formula. I'm looking to solve the wave equation, subject to these same ...
2
votes
0answers
57 views

solving this differential equation for $y$, Is it even possible?

Lets say I have the following: \begin{gather} \frac{(y')^3 + 3 y' y'' + y'''}{(y')^2 + y''} = \sqrt{1+(y')^2}\\ \frac{((y')^3+3y' y'' + y''')^2}{((y')^2 + y'')^2} = 1+(y')^2\\ \frac{(y')^6 + 6 (y')^4 ...
2
votes
0answers
36 views

Initial conditions to solve an ODE?

Given is the following inhomogenous linear ODE (4th order): $$q_0\cdot\sigma + q_1\cdot \dot\sigma + q_2\cdot \ddot\sigma + q_3\cdot \dddot\sigma + q_4\cdot\ddddot\sigma = p_0\cdot\epsilon + ...
2
votes
0answers
45 views

Solving $2y'''(t)+3t\ y(t)=0$.

For a certain problem, I am trying to solve the ODE $$2y'''(t)+3t\ y(t)=0$$ I am pretty clueless what to do here, any hint would be appreciated. Thank you very much.
2
votes
0answers
30 views

Differential Equation. VERY small problem

I want to ask a question later, after I show you this TESTING: x^2 = 1 Differentiate both sides 2x = 0 TESTING: x = 1 Differentiate both sides dx/dx = 0 1 = 0 So when can I differentiate both ...
2
votes
0answers
25 views

How to find Laplace transform of a differential equation?

$y′′ + 3y′ + 2y = f$ , $y(0) = 0$ , $y′(0) = 1$ where $f$ is given by $f(t) = \sum_{n=1}^\infty \delta(t−n)$; find a 1-periodic function $y_*$ with $\lim_{t\rightarrow \infty} |y(t)−y_*(t)| = 0$. I ...
2
votes
0answers
74 views

How to solve system of stochastic differential equations?

I have the following two SDEs $$dN_1=(2a-1)pN_1dt+\alpha_1 N_1dW_1$$ $$dN_2=(2pN_1-\mu N_2)dt+\alpha_2 N_2dW_2$$ $W$ is the standard Brownian motion/Weiner process. This isn't homework, I'm just ...
2
votes
0answers
119 views

Lyapunov function for a damped pendulum

The question is about damped pendulum. There are two statements I don't understand or I'm not sure if my justification for them is correct. Could you say if I'm right? The example is from a German ...
2
votes
0answers
82 views

Second order differential equation, power series method

Solve the differential equation $$(x+2)y''-xy'+(1-x^2)y=0 ; \quad X_0=1$$ using the power series method about the point $x_0=1$. I get to this step after deriving the derivatives of the ...
2
votes
0answers
18 views

How is it possible to continue solutions for a differential equation along t?

Given the equation $y' = e^{\sin y+t} + t\cos(y)$. I rewrote it as $$ y(t) = y(0) + \int_{0}^{t}ye^{\sin y+t}+t\cos y $$ I'm asked to prove that every solution can be continued for every t. I know ...
2
votes
0answers
72 views

Study of a system of differential equations

I'm asked to study everything that is possible to know about the sytem$$\begin{cases}x'=x^2-y^2\\y'=2xy\\z'=-z\end{cases}$$ My questions here is, how much can be know about it?, how do I know I ...
2
votes
0answers
43 views

Find an integrating factor such that $y'=\frac{1-x+y}{x-y}$ is exact

Yet another question of this sort, and hopefully the last. In the previous question I posted, we were lucky enough and the integrating factor was a function of only one variable, the ansatz $\mu_y=0$ ...
2
votes
0answers
58 views

Annoying differential equation involving composition

Upon trying to crack into a problem, I managed to end up with the following differential equation. $$ y = xy' - y'\circ y', \qquad\text{or}\qquad y(x) = x\cdot y'(x) - y'(y'(x)) $$ I haven't a clue ...
2
votes
0answers
67 views

Solution of differential equation $x'=Ax$ where $A=PJP^{-1}$

Let $A$ be a $n\times n$ matrix. Suppose $A=PJP^{-1}$ being $J$ the Jordan form of $A$. Prove that if $x(t)=(x_1(t),\dots,x_n(t))$ is a solution for $x'=Ax$, then ...
2
votes
0answers
36 views

Find the 3rd order DE whose general solution is $ y= C_1e^{2x} + C_2\cos x + C_3 x\sin x $

My attempt $$ \begin{matrix} y &=& C_1e^{2x} &+& C_2\cos x &+& C_3x\sin x\\ y' &=& 2C_1e^{2x} &-& C_2\sin x &+ &C_3(\sin x &+& x\cos x)\\ ...
2
votes
0answers
36 views

A basic ODE question

Let $G \subset \Bbb R^d$ be open and let $ V: G \to [0, \infty)$ be such that $\dot{V} = \nabla V.h : G \to \Bbb R$is non-positive. We assume that $H=\{x: V(x) =0\}$ is equal to the set $\{x: ...
2
votes
0answers
163 views

Famous parametric curves that are solutions to differential equations

I know that the cycloid satisfies the differential equation $ \left( \frac{dy}{dx} \right)^2 - \frac{2r}{y} + 1 = 0. $ Are there other famous plane curves that are also solutions to a differential ...