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

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Get a special form of an linear System of ODE (using polar form)

In this post Converting an ODE in polar form it is shown that a linear system of ODE $$ x'=\begin{pmatrix}a(t) & b(t)\\c(t) & d(t)\end{pmatrix}x $$ can be written in polar coordinates ...
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Prove $x\to 0$ as $t\to \infty$ if we consider the system of equations $x'=(A+B(t))x$ where $B(t)\to 0$ and $A$ has negative eigenvalues.

Consider a matrix $A$ such that all of its eigenvalues are negative. Consider $B(t)$ where $B(t)\to 0$ as $t\to\infty$. Then consider the system of equations $$ x'=(A+B(t))x$$ Prove that any ...
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32 views

Limit of solution of linear system of ODEs as $t\to \infty$

I am completely stuck on the following problem: Consider the linear system: $x'(t)=A(t)x(t)$ where $A(t)$ is an $n$ by $n$ matrix. Assume that $\lim_{t\to \infty}A(t)=B$. Suppose that each eigenvalue ...
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Solving ODE involving matrices

We have a given ODE $ K(x)_{_{3 \times 3}}=xC_1K(x)+x^3C_2K'(x) \tag 1$ where $C_1,C_2$ are constant skew symmetric matrices of dimension $3 \times 3$ with determinant $0$. How do we solve ...
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Periodic solutions and critical points

I was going through a lecture, and for an ODE: $x' = x(5-x-2y), y'=y(-6x+x+3y)$ Which has critical points at : $(0,0) (0,2) (3,1) (5,0)$ My professor posed the question as to why the periodic ...
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1-forms and zero simple

Let $\varphi$ a differential transformation such that $\varphi (x,y)=(f(x,y),g(x,y))$ and $D\subset U$ such that $\varphi$ restricted to $\partial D=\gamma$ be distinct zero and we define $i(\varphi ...
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21 views

The Adjoint Equation

I have a simple question about the adjoint equation for second order linear differential equation. Given an equation of the form $$P(t)y'' + Q(t)y' + R(t)y = 0$$ Let $u(t)$ be an integrating factor ...
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Control Function with solution and fixed initial data on time interval, critical point of a cost functional?

Let $u(t)$ be a solution of the ODE $u''(t)+tu'(t) + u(t) = f(t)$ on the time interval $[0,T]$, with fixed initial data $u(0)=u_0$, $u'(0) = u_1$ where $f(t)$ is a control function. Find $f(T), ...
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A question on First Order Differential Equations

So, I've recently begun to tutor friends in math. I've only tutored classes that I've taken [algebra-multivariable calculus], and last night I was tutoring a friend in calc II. He pulled out a take ...
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How to estimate the local error and the global error for Runge-Kutta method

How to estimate the local error and the global error for Runge-Kutta method in practice? I have no idea. I recieved a nice answer on the question at other site
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Best approach to matrix representation of system of nonlinear ODEs

I have this system of ODEs: $$ \frac{dS}{dt}=\pi S-\beta S Z\\ \frac{dZ}{dt}=\alpha S Z - \delta Z $$ I am trying to rewrite them in the form : $$ \pmatrix{\dot{S}\\\dot{Z}}=\mbox{diag}(S,Z) ...
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Simplifing a Cauchy product to find the recurrence relation when solving a differential equation using a power series solution.

I'm having trouble finding the recurrence relation of the following non linear differential equation: $y''(x)+p(x)y'(x)+y^2(x)=0$ with $y(0)=1$ and $y'(0)=0$ where ...
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Is there a relationship between the integrand in Green's Theorem and the test for finding an integrating factor for a differential form?

Green's Theorem has the formula $$ \int_C Mdx+Ndy=\int\int_D\left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right)dxdy $$ There is also a well known test for finding an integrating ...
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How to solve a system of two differential equations describing the concentration in a leaky tank?

While filling up a chemicals container at a constant rate of 300 litres/min, the crew of a naval ship discover two leakages at the bottom of the container. They discover that the chemical is leaking ...
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Second Order Differential Equations - Undetermined Coefficients

When solving for this one: $y''-3y'-4y=e^{-x}$ For the trial function, let: $y=Ae^{-x}$ $y'=-Ae^{-x}$ $y''=Ae^{-x}$ $=> Ae^{-x}-3(-Ae^{-x})-4(Ae^{-x})=e^{-x}$ $=> ...
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List of eigenvalues for the Schrödinger equation

I'm writing an algorithm which computes the eigenvalues $E$ of the Schrödinger equation with potential $V(x) = x^2$, ie the harmonic oscillator. The equation is defined as follows $$ y''(x) = ...
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Solving a system of 3 DE's. Need a tip for finding eigenvecotr

Hello I have this system: $$ x'=2kx + ky + kz, y'=kx+ 2ky + kz, z'=kx+ ky+ 2kz $$. I found that λ=k and 4k. I am solving for the first eigenvector when λ=k, and end up with this: a+b+c=0, can anyone ...
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32 views

Mixing Problem, Different Flow Rates

This is a double question and the first part reads Consider a tank holding 100 gallons of water in which are dissolved 50 pounds of salt. Suppose that 2 gallons of brine, each containing 3 pounds of ...
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Infinitesiman generator of Time dipendent process

I'm trying to find the infinitesiman generator of this process $dY_{t}=\dfrac{b-Y_{t}}{1-t}dt+dB_{t}$ $0\leq a <1$, $Y_{0}=a$ where $B_{t}$ is a brownian motion; and I've found the solution: ...
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How do I determine sufficient conditions for the existence of the solution of an initial value problem?

Suppose that $f$ is a smooth function from $\mathbb R^{3}$ to $\mathbb R$ with $f(0,0,0)=0$. Under what sufficient condition will the differential equation $f(x,y,y')$ have a solution satisfying the ...
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63 views

Euler method inequality

Given the problem for $t\neq0$ and $t\le1$ $y'(t)=y^2(t)$ $y(0)=1$ Let $\mu>0$, and $\epsilon_n=\frac12(f(t_{n+1},y_{n+1})-f(t_n,y_n))$, such that $|\epsilon_n|\le\mu|y_n|$ is ...
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14 views

matrix in normal coordinates

Writing the matrix $ \begin{pmatrix} -\frac{k}{\gamma} & \frac{k}{\gamma}&0&0&0&\cdots&0&0&0&0 \\ \frac{k}{\gamma} &-2\frac{k}{\gamma}& ...
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48 views

Hopf bifurcation how to prove

I have this system of differential equations: \begin{equation} \frac{dx}{dt}=1-(b+1)x+x^2 y\\ \frac{dy}{dt}=bx-x^2 y \end{equation} I now that we will have a bifurcation when $b$ grows and ...
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Determining the Approximate Solution to an ODE

This is a homework question I am stuck on. Say $\varepsilon \in (0, 1]$, and consider the IVP $$ x'(t) = -3x(t) + \varepsilon x^2(t) \ln (x(t)), \quad x(0) = \frac{1 + \sin(\varepsilon)}{2}. $$ If we ...
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25 views

On Yau's (and Schoen's) proof of the positive mass theorem

I would like to face the proof of the positive mass theorem by Yau and Schoen. I have a Bsc in Mathematics and a Msc in Theoretical Physics and I'm preparing a PhD interview-challenge where I have to ...
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22 views

Discontinuous Linear Differential Equation

Consider the IVP $$ y' +p(t)y(t) = 0$$ $$y(0) = 1$$ $\forall\,t\in[0,1],\quad p(t)= 2$ $\forall\,t>1,\quad p(t)= 1$ Solve this IVP by first solving for t ∈ [0, 1] and then for t > 1 to obtain ...
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Linearizing a DDE about a point

I want to show that the delayed logistic equation $$ N'(t)= rN(t) \left[1-\frac{1}{K} \int_0^\infty N(t-u)k(u) \> {du}\right]$$ where $$k(u)= \frac{1}{\tau} \exp\left(\frac{-u}{\tau}\right)$$ ...
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Lyapunov stability in nonlinear system

Please, is there any reference for proposition below or does it perhaps follow from a standard fact? I've got it for some other problem but I actually do not know how to comment it in my article. ...
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A program for draw differential equations

I need to analyze the autonomous equation $ \ddot x +x +\alpha x^2 = 0 $ I need to analyze its flow about its critical points, can you suggest me a program in which I can draw this equation ( $\dot x ...
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Find two linearly independent solutions. Use Frobenius method to solve this equation.

$\displaystyle 2xy'' + (1-2x^2)y' - 4xy = 0$. This is what i got when i solved for the recurrence relation: $(2(n+r)(n+r+1) + (n+r+1) )c_{n+1} =( 2n+2r+2)c_{n-1} $ $\displaystyle c_{n+1} = ...
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Proving equivalence of equations

I need to prove the equivalence of two equations, and have reduced them to: $f'''+ff''+\beta(1-f'^2)=0$ Eq(1) and $f'''+\frac{1}{2}(m+1)ff''+m(1-f'^2)=0$ Eq(2) with the ...
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50 views

Ode with step function in the right-hand-side

I want to solve the following ODE: $$\dot{X}(t,x)=F(X(t,x))$$ where $F(x)=1$ if $x>0$ and $-1$ if $x<0$. How to treat this discontinuous right-hand-side?
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difference between runge kutta methods of same order

I recently read about runge kutta methods for solving differential equations. So far I understood the idea but up to know nobody could answer me following question: If we consider the explicit rk ...
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Application of Sturm comparison theorem

Let $w(x), q(x)$ be continuous functions on $[a,b]$ and $q(x) < 0, \forall x \in [a,b].$ Could anyone advise me how to use Sturm comparison thm to show any non trivial solution to $y^{\prime\prime} ...
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How to solve this Ricatti-like ODE

I have been trying to solve the following ODE \begin{equation*} \dfrac{d\pi}{dx}x=c_1+\pi(x) c_2 + \pi(x)^2(c_3-x), \end{equation*} where, for every $i=1,2,3$, $c_i$ is a constant real value. ...
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Differentiation of an integral with respect to time variable

I have a little doubt about an integral that I try to differentiate according to time $t$. I have tried to do it with Leibniz rule but it did not work. Here it is ; ...
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Convergence order of Runge-Kutta methods: proof requested

I have been told that: The convergence order of an explicit Runge-Kutta method with $s$ stages is at most $s$. Furthermore, for $s>5$ there is no explicit Runge-Kutta $s$-stage method of order ...
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Runge-Kutta methods and butcher tableau

What does the Butcher tableau of a Runge-Kutta method tell me about the method, besides the coefficients in its formulation? In particular, what requirements about it guarantee consistency and ...
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Differential equation with $x'(t)=\sqrt[5]{(x(t))}$

Solve the following Cauchy problem for $t\in\mathbb R$ $x'(t)=\sqrt[5]{(x(t))}$ $x(0)=0$ Is the solution unique ? Now this is a differential equation of the form $x'=f(x)$, thus; ...
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Solution to non-linear OIDE

How do I go about solving this equation? $\frac{\partial F(r,y)}{\partial r} = Q(r,y) - P(r,y) F(r,y) - R(r,y)F(r,y)\int_0^\infty dy'S(r,y') F(r,y')$ with the initial condition that $F(r=0,y) = 0 \ ...
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Explicit solution to a first order nonlinear ODE

Is there any explicit solution to the following ODE? $G'(z) =aG(z)+bG(z)^α-c$ $G(0) = d_0 $
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Asymptotic behaviour of $\varphi''(x)=F(\varphi(x))$

I'm concerned with the discussion of a ODE, especially the discussion of the solution. I've got the assumptions that there is the relation $\varphi''(x)=F(\varphi(x))$ for all $x$ on $\mathbb{R}$. ...
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homogeneous BVP has at most one linearly independent solution

I am trying to understand following proof. I understand the set up however can't make the connection with the Picard Lindelöf Theorem. Can you please help me with this? Statement: The homogeneous ...
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Differential Equation Find general solution of y'' - y =cosh(x) using variation of parameters

Hello I am having some issues with the simplification of the DE, I am okay up on till $$y_p(x)=v_1(x)y_{p1}(x) + v_2(x)y_{p2}(x) $$ $$ \frac12 e^{-x}\left(\frac {-e^{2x}}4-\frac x2\right)+ \frac ...
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$M(x,y)\,dx+N(x,y)\,dy=0$ can always be converted to the form $dy/dx=F(y/x)$

I'm being asked to prove that all $M(x,y)\,dx+N(x,y)\,dy=0$ can always be converted to the form $dy/dx=F(y/x)$. My guess is that I just have to manipulate and that's easy. But I have no clue what ...
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How do you draw a phase potrait plot when one of the eigenvalues is zero?

So I have the system $X'=\begin{pmatrix}a&1\\2a&2\end{pmatrix}X$ I found the eigenvalues to be $\lambda=(a+2,0)$ and the eigenvectors to be ...
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30 views

Describe as a system of differential equations

How is it possible to describe the $\dfrac{du_c}{dt}$ and $\dfrac{di}{dt}$ as a system of differential-equations? $$ u_s = u_r + u_l + u_c\\ i = C\frac{du_c}{dt}\\ u_l = L\frac{di}{dt}\\ u_r = Ri\\ ...
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Proving that maximal interval of existence exists and that solution is unque

For each $\lambda\in \mathbb{R}$, let $\varphi_{\lambda}$ : $J_{\lambda}\rightarrow \mathbb{R}$ denote the solution to the following initial value problem: $$ ...
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97 views

Qualitative properties of solutions to a ordinary differential equation.

I have this problem : $$\begin{cases} -(p(t)u'(t))'=f(t,u(t))\\u(0)=u(+\infty)=0\end{cases}$$ we have that $u$ is continues, $f:\mathbb{R}^+\times \mathbb{R}\rightarrow \mathbb{R}$ is continuous and ...
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Matrix linearization of the Lagrangian points.

I have to solve a long problem, and I´m in trouble in a step. The step is to linearize the next differential equation, by writtin its correspondient Jacobian, and then, calculate the eigenvalues of ...