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

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Does a Liapunov function h to have all the variables explicitly?

If I have for example, a system like this $$\begin{matrix} \dot{x}=f(x,y) & \\ \dot{y}=g(x,y) & \end{matrix}$$ in which i have to prove stability using a Lyapunov function. Now, if i have ...
2
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1answer
24 views

A simple problem on first order differential equations

An ODE (Ordinary Differential Equation) of order $n$ becomes a relation: $$F(x,y,y^{(1)},...,y^{(n)})=0$$ Then $F(x,y,y^{(1)})=0$ defines an ODE of order one. In "basic standard texts", for purposes ...
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15 views

perturbation of exponentiolly stable system

consider the following system on $\Bbb{R}^n$ $\dot{x} = f(x,t)+g(x,t) $$ $$ $$ $ $ (*) $ assume that f(0,t)=g(0,t) = 0 and 1. 0 is an exponentiolly stable equilibrium of $\dot{x}=f(x,t)$ ...
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1answer
22 views

First Order Differential Equation for a Harmonic Oscillator

A box with mass $m$ is attached to a spring with spring coefficient $k$. This system is then placed into a glass case filled with a liquid with drag coefficient $\alpha$. Now I have the following ...
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2answers
21 views

Differential equation$ (x^2-x)y' = (y^2+y)$

Can i get help solving the differential equation $$y' = \frac{y^2+y }{x^2 -x}$$ I tried searching but could not find anything similar. Thank you!
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1answer
10 views

Maximally extended solution of this ODE.

So I am asked to find the positive, maximally extended solutions to this ODE. $$u'(x) = \frac{x}{u(x)}$$ Now a solution is given by $$u(x) = (\int_{y_0}^y t dt )^{-1}\circ \int_{x_0}^x s ds = ...
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16 views

Solve $A \partial_t w + B \partial_t\partial_x^4 w + C \partial_x^4 w + \partial_t^2 w = 0$

a non-mathematician wants me to solve a PDE. The problem is that I don't know a lot of theory to solve PDE's except the fouriertransform. This is the PDE $$A \partial_t w + B \partial_t\partial_x^4 w ...
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1answer
17 views

Approaches to stability of newtonian systems

I am having some difficulties figuring out how to approach "Test stability problems". I usually test the linearization of the system (since it is very straightforward and easy), and if that doesn't ...
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1answer
27 views

When does the same trajectory appear in two dynamic systems from the same point?

Imagine you have two dynamical systems, given by the statespace equations: $\frac{dx}{dt}=F_1(x)$ and $\frac{dx}{dt}=F_2(x)$, and you are concerned with trajectories form a point in phase space $x_0$. ...
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3answers
28 views

Differential equation $2f'(t)+tf(t)=0$ with $f(0)=\sqrt{\pi}$.

How to solve the following differential equation: $$2f'(t)+tf(t)=0$$ with $f(0)=\sqrt{\pi}$. I tried to write $2f'(t)+tf(t)=0$ something like $(f(t)g(t))'=0$ for some function $g$ but it was ...
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1answer
10 views

Expressing $x$ and $z$ as functions of $y$ (non-generate matrix)

Consider the system $$ \dot{x}=x-z+y^2,\quad\dot{y}=x-2y+z+y^2+2x^2z,\quad\dot{z}=-2x+2y+z^2-y^2. $$ and the equilibrium $(0,0,0)$. Now, there is used some statement that I did not know yet: ...
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1answer
31 views

Prove that if $A\neq B$ then $\exp(A/n) \neq \exp(B/n)$ for some $n\in \mathbb N$

Let $A \neq B \in M_{n\times n}$ be linear maps. I'd like to prove that there exists $n\in \mathbb N$ such that $e^{A/n} \neq e^{B/n}$. I tried assuming that $e^{\frac{A}{n}} = e^{\frac{B}{n}}$ for ...
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1answer
25 views

Why is this system reversible? What does this mean?

Consider the system $$ \dot{x}=y,\qquad\dot{y}=-x+y^2. $$ Then, it is said that the system is reversible $(t\to -t, y\to -y)$. What does this mean? If I put this into the equations, I get $$ ...
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1answer
31 views

How to prove $\frac{dh}{dt}= \frac{5 }{h^2} - \frac{1}{20}$ and a couple other related questions (complete information inside)?

This is a differential equation question. Since I might not be able to explain is well, I will attach a link to the question as well as a screenshot of the mark scheme. Question: Mark Scheme: ...
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0answers
18 views

Reachability from non-zero initial state?

I have the following system: $$ \dot x(t)=\begin{bmatrix} -2 & 1 & 2 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}x(t)+\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}u(t) $$ The ...
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10 views

capillary surface problem [on hold]

Consider the capillary surface problem (⋆) )   Du  div 1 + |Du|2 = κu in Ω on∂Ω,  Dηu  1+|Du|2 =β where κ > 0, η is the outward pointing unit normal to ∂Ω and β ∈ C1(Ω) satisfies |β| ≤ 1 ...
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16 views

In a recurrence relation, how do we know which order to terminate?

By employing Frobinious or Power Series approach, we my come up with a recurrence relation that is only solvable if we set any constant lower than $a_0$ or higher than $a_n$ vanish. For example, in ...
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1answer
17 views

checking that an initial condition holds for the heat equation

I'm trying to follow a video lecture on solving the heat equation. $I) \space u_t = ku_{xx}, x \in \mathbb{R}, t > 0$ $II) \space u(x,0)=\phi (x), $ $k$ is const, $\phi (x) $ is a ...
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1answer
25 views

Wolfram Alpha Step By Step For Systems of differential equation

Does anyone know if wolfram alpha has step by step solutions for systems of differential equations? When I input them, it comes up with an answer but it does not give me the step by step solution. I ...
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1answer
33 views

Raising e to the power of both sides of an equation

I have a simple question: in differential equations, it has been common in several of my homework problems to raise a base $e$ to the power of both sides of an equation to get variables out of natural ...
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1answer
17 views

Differential equation with shifted argument.

What are the methods for solving the following class of problems: $$ \frac{df(x)}{dx}=a f(x-\xi), $$ or $$\begin{cases} \frac{\partial F(x_1,x_2)}{\partial x_1}=a_1F(x_1-\xi_{11},x_2-\xi_{12})\\ ...
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1answer
28 views

Help with this differential equation, nonlinear

How would I solve the following Differential Equation $\frac{dy}{dx}= \sqrt{x+y} $ Clearly, it is nonlinear and non homogeneous, I could not find the way to solve it with Bernoulli or to make it an ...
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0answers
21 views

Repeated Eigenvector/Eigenvalue matrix method

So I am having trouble with finding the generalized solution and I am not sure why my answer is interpreted as incorrect and I wanted to double check. $$ \overrightarrow{y'} = \begin{pmatrix} -6 ...
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1answer
26 views

Trapezoidal rule - truncation error

I am trying to prove that when solving numerically diff. eq.: $$ y'(t)=f(t,y(t)), \hspace{0.5cm} y(t_{0})=y_{0} $$ using trapezoidal rule, namely: $$ y_{n+1}=y_{n} + \frac{h}{2} \left( f(t_{n},y_{n}) ...
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0answers
22 views

When are solutions of nonlinear ODE bounded?

I wonder if there is a method other than solving to find out whether the solutions of a specific differential equations are bounded? I am interested in equations of the form $$ \frac{dx_i(t)}{dt} = ...
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2answers
50 views

Derivative of improper integral.

Having trouble trying to differentiate this. $y(t)=e^{it} + \alpha\int_{t}^{\infty} sin(t-s)\frac{y(s)}{s^2} ds $ ...
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1answer
23 views

Soft question about connection between flow and group actions

I am learning about flow and came across the group action formal definition of flow on Wikipedia. First of all, why is it a group action of the real numbers on the set of particle positions? Is this ...
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2answers
26 views

Emden‐Fowler differential equation

Good day. I am trying to solve the following equation: $$\ddot{y}(x)-\frac{A}{x}\dot{y}(x)+\frac{Bx^2}{2}y(x)=0.$$ WolframAlpha says it is an Emden‐Fowler equation, but I have no idea how to solve ...
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0answers
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Proof that different ways of approaching Green's Function gives the same answer

I am learning about Green's function, and the different ways of obtaining it, through variation of parameters and through eigenfunction expansion. Variation of parameters gives it via one equation ...
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1answer
37 views

Solving $(4y+2x-5)dx+(6y+4x-1)dy=0$ using 2 methods produced 2 different answers!

$$(4y+2x-5)dx+(6y+4x-1)dy=0,y(-1)=2$$ First method: $$\frac{dy}{dx}=-\frac{4y+2x-5}{6y+4x-1}$$ let $Y=y-\frac{9}{2}$, $dY=dy$ and $X=x+\frac{13}{2}$, $dX=dx$; ...
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1answer
24 views

Analytic solution to a second order nonlinear ODE involving $\operatorname{sech}^2(x)$?

I am trying to look for an analytic solution to the following equation $\frac{1}{4D} (y')^2 -\frac{1}{2}y'' = -n(n+1)A\operatorname{sech}^2(\frac{x}{b})$ with $A>0$, $D>0$ and $b>0$ and ...
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1answer
43 views

Solve $(x+y)^2dx+(2xy+x^2-1)dy=0$

I have a problem trying solving this ODE: $$(x+y)^2dx+(2xy+x^2-1)dy=0$$ I tried the following s=teps: $$M=(x+y)^2dx, N=(2xy+x^2-1)dy$$ $$\frac{\partial M}{\partial y}=2(x+y)=2x+2y$$ $$\frac{\partial ...
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1answer
24 views

Solve ODE of the form $X(t)+Y(t)X(t)'=G(t)$ for $X(t)$

I have this ode as the following, $$\lambda(t)^\prime+p(t)\lambda(t)=G(t)$$ I need to solve for $\lambda(t)$. Any suggestion or hint is much appreciated.
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31 views

The derivative of a function of multiple variables

I am trying to understand a step in the theory section of my differential equations textbook. The author writes, For example, suppose we transform the first order differential equation ...
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2answers
23 views

ODE problem, now a days asked quite frequently (inhomogeneous)

How to solve $$\frac {ds}{dt}+s=|t|,~s(0)=1.$$ Because i have never seen before such a inhomogeneous ODE, where on right hand side there is modulus function. What i did to solve it, i broken up ...
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2answers
79 views

Why does the solution of $y'' = -(1+e^x)y$ decay like $e^{-x/4}$?

I am trying to show that any solution of $y'' = -(1+e^x)y$ goes to $0$ as $x \to \infty$. I was able to show that the solutions hit $0$ infinitely often. It seems (empirically) that the solutions ...
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0answers
19 views

how to solve this differential equation in matrix form?

I am reading the article [1] and came across a differential function that the authors did not provide detailed steps of the solution. The function is of great interest to me. Any thoughts on how the ...
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3answers
22 views

Solution to Nonlinear, Autonomous ODE with square root?

So I have been unable to solve the following equation: $y'= \sqrt{1-y^2}$, $\:\:y(0)=-1$ I have tried z-substitution to no avail, and was wondering if anyone had any ideas on how I should go about ...
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1answer
25 views

Analytical expression of the solution to parabolic equation

What is the analytical expression of the solution of the following 1D parabolic equation ? $$\dfrac{∂f}{∂t}+V\dfrac{∂f}{∂x}=λ$$ where $t, x$ – independent variable (time and space position ...
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3answers
87 views

Is $(-\infty, 0)$ the same size as $(0, \infty)$?

A differential equations problem asked about the largest interval on which the solution was defined. The solution was defined except for $t=0$, which made me wonder whether the intervals $(-\infty, ...
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2answers
21 views

Fixed points for 1-D ODE

I'm doing some independent work, and have managed to come across the following interesting 1-D autonomous ODE: $\dot{x} = x(1-x) \log^2\left[\frac{x}{1-x}\right]$. For the fixed points, i.e., where ...
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1answer
36 views

show the Wronskian is constant

Let $p,q : \Bbb{R} \to \Bbb{R^n}$ and $H:\Bbb{R^n}\times \Bbb{R^n} \to \Bbb{R} $ and the hamiltonian system: $$ \begin{cases} \dot p = - \frac{\partial H}{\partial q} \\ \dot q = \frac{\partial ...
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1answer
45 views

How to use Fourier's transform to solve differential equation

I have to solve following problem: $$ u_t(t,x) = \Delta u(t,x) $$ $$ u(0,x) = f(x) $$ I've started: $$ \frac{\delta}{\delta t} F(u(t,\xi))=F(u_t(t,\xi))=F(\Delta u(t,x))$$ and here I've stoped, ...
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27 views

rayleigh quotient of eigenvalue problem (sturm liouville theory and partial differential equations)

I am reading "A First Course in Partial Differential Equations with Complex Variables and Transform Methods" (Weinberger, p. 168). if we have the eigenvalue problem $$ (pu')'- qu + \lambda \rho u = 0 ...
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1answer
21 views

Derivation of the variation of parameters in Second-Order Differential Eq.

In Second-Order ODEs ,There is a problem which I haven't solved. Method of Variation of Parameters; In derivation of the method , there is a part which is following ; P and Q just constant. ...
0
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1answer
51 views

Hermitian Operator? Proof by complex eigenvalue

Is the operator $\frac{1}{x} \frac{\partial }{\partial x}$ Hermitian? I think it is because I found $f(x)=Ce^{\frac{1}{2}x^{2}\lambda}$ to be the set of eigenfuctions with ${\lambda}$ as the ...
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22 views

Math - Differential Equations [on hold]

A differential Equation: a) can always be solved explicitly b) can always be solved implicitly c) can have no solution d) can have imaginary solutions e) can have infinite order
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20 views

Picard theorem for ODE in complex variable

Given a ODE in real variable, the well-known Picard theorem guarantees the existence and uniqueness of the solution. Does there exists a counterpart for ODE in complex variable. I am looking for a ...
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1answer
37 views

Construct differential equation given the phase portrait (non-linear pendulum)

I am curious to know how to recover the differential equation that goes with a phase portrait. I have seen the following posts but the first one was a $y'$ (and easy enough to "do in my head") and ...
0
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1answer
29 views

How to show the solution is unique to ODE given initial value?

I am new to the ODE concepts, and I am a bit confused how to prove a seemingly simple question: If $x'=kx$, where $x(0)=5$ and $k=-10$, why is $x(t)=5e^{-10t}$ the unique solution under such ...