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

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41 views

About the boundary conditions of the Black-Scholes-Merton PDE

I have a question about the solution of the Black-Scholes PDE for the European call option when I read the book Stochastic Calculus for Finance II of Steven E.Shreve. Let $c(t,x)$ be the value of the ...
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0answers
38 views

Functional equation + differential equation = way of finding solution?

Question I was wondering about the following: Let's say there is a differential equation whose solution is $f$ And $f$ also satisfies a functional equation. Can anyone construct an (non-trivial) ...
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3answers
50 views

Separating variables by substitution in a homogenous ODE

I am brand new to ODE's, and have been having difficulties with this practice problem. Find a 1-parameter solution to the homogenous ODE:$$2xy \, dx+(x^2+y^2) \, dy = 0$$assuming the coefficient of ...
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1answer
255 views

Lyapunov equation for stability analysis - what's the point?

Straight from Wikipedia: In the following theorem $A, P, Q \in \mathbb{R}^{n \times n}$, and $P$ and $Q$ are symmetric. The notation $P>0$ means that the matrix $P$ is positive definite. ...
3
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1answer
54 views

What does a 3D periodic solution of a differential equation look like?

The Pointcare-Bendixson Theorem implies that if a solution stays in a bounded region with no equilibrium points then it is either a periodic solution or it approaches a periodic orbit as t goes to ...
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0answers
41 views

Theorem to show trajectories of differential equations are close after small change to initial condition

Consider two solutions(or trajectories), say $x_1(t)$ and $x_2(t)$, of a system of differential equaions. That is, $$ x_1'(t)=x_2'(t)=f(x,t), t\ge0. $$ Also, $\|x_2(0)-x_1(0)\|<\epsilon$ for some ...
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3answers
337 views

How to interpret complex eigenvectors of the Jacobian matrix of a (linear) dynamical system?

Consider a linear ODE system of the following form: $$ \frac {dx} {dt} = Ax $$ In case $A$ has real eigenvectors, I can interpret them as the directions in which the system will move, if the initial ...
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0answers
27 views

Does an exponential bound on a Lyapunov candidate imply asymptotic stability?

If I have a Lyapunov candidate $V:[0,\infty)\rightarrow \mathbb{R}$ and I'm able to show that $$ V(t)\le k e^{-\eta t} V(0),\qquad \forall t\in[0,\infty) $$ can I conclude something about ...
21
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2answers
412 views

Periodic orbits of “even” perturbations of the differential system $x'=-y$, $y'=x$

Fix some even functions $f$ and $g$, differentiable, such that $f(0)=g(0)=0$ and $f'(0)=g'(0)=0$, and consider the autonomous differential system $$\left\{\ \begin{array}{lcr}x'&=&-y+f(x)\\ ...
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1answer
63 views

What is a critical point in a system of equations?

I have an assignment question based around a system of nonlinear differential equations, $$ x' = f(x, y) \\ y ' = g(x, y) $$ The first part of the question is to locate and classify all the ...
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1answer
82 views

Runge Kutta stability

I am facing a problem solving a ODE with a Runge-Kutta 4th order method: The expression in order to solve is : \begin{equation} Ay^{''}+By^{'}+Cy= Cu \end{equation} \begin{equation} y =OUTPUT ...
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1answer
38 views

Flow of time-depended vector field

Suppose $X_t$ is a time-depended vector field with flow $\phi_t$, so, $\frac{d}{dt} \phi_t = X_t(\phi_t)$. Is it true that $d \phi_t(X_t(x)) = X_t(\phi_t(x))?$ This is true when $X_t$ does not ...
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1answer
87 views

Units of ODE solution don't match

I have to solve the differential equation: $v\,'=g-cv$. Sorry in advance for lack of latex. I will learn it soon, please let me make a question using the common programming notation for my ...
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1answer
39 views

Help with Euler Equations

This is from my textbook. Can someone give me a better explanation of what to do here? What does part (a) mean, i.e., how am I supposed to write $x = ln(t)$ in terms of $\frac{dy}{dx}$ and ...
4
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1answer
66 views

When can you take the limit of a parameter before solving the differential equation?

Short example: consider the differential equation \begin{align*} f'(x)=\frac{k^2}{k^2+k+1}xf(x) \end{align*} where $k$ is a parameter. Wolfram Alpha tells me that the solution to this equation is ...
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257 views

How to classify/ solve this PDE?

I am searching how to solve the PDE below but I can not seem to find a decent example online. My major did not focus much in solving PDEs so I feel very deficient. I know how to solve for the steady ...
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1answer
180 views

Flow of sum of non-commuting vector fields

Let $V,W\in\Gamma(M)$ be any two vector fields. Is there any "nice" expression for the flow of $V+W$ in terms of the flow of $V$ and the flow of $W$? It would be sufficient for me to have some sort of ...
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3answers
45 views

Forming differential equation

I'm trying to get from: $$e^{\lambda t} (\frac{dN}{dt} + \lambda N) = re^{\lambda t} $$ To: $$ \frac {d}{dt}(Ne^{\lambda t}) = re^{\lambda t} $$ However I'm not sure what procedure to use to go ...
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2answers
42 views

question on second -> first order systems [duplicate]

I have heard that it is possible to write second order IVP as first order system. What are some strategies to writing $y''=xy^2$, $y(0)=1$, $y'(0)=2$ as a first order system $y'=f(y)$, $y(a)=y_0$? ...
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92 views

Lyapunov stability at origin with identically zero test function

At the origin, determine stability of $$x' = y \\ y' = -\tan(x)$$ If we use the test function $V(x,y) = 0.5y^2 + \int_0^x tan(s)ds$, we get $\dot{V}=x'\tan x +y'y = y\tan x -y\tan x = 0$, so the ...
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0answers
59 views

Separation of variables, Homogeneous or Exact Differential equations?

So I've just encountered these three, during exams of course they don't tell you which one is to use, if you need to use separation or homogeneous or exact. I was just wondering is there like a ...
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38 views

The number of characteristic curves of a pde

When a partial differential equation is elliptic, $B^2-4AC\lt 0$ and eigenvalues are complex. does there exist any characteristic curves?
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230 views

Simple related rates derivative question

Rafael is walking away from a $12$-ft-tall lantern at a constant speed. If the tip of Rafael's shadow is moving twice as fast as he walks, how tall is Rafael? I'm confused on the step where $dL/dt = ...
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1answer
28 views

Finding a power series solution for a given differential equation and identifying the function represented by the power series.

Find a power series for the solution of the differential equation $y'(t)-2y(t)=0 ,\ y(0)=5$, and then identify the function represented by the power series. (I use the following information ...
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2answers
44 views

Euler Cauchy equations, change of variables

To convert an euler cauchy: $x^{2}y''+pxy'+qy=0$ equation into a linear one we perfom the substitution $x = e^z$ from which we get: $$z=\log x$$ $$\frac{\mathrm{d} x}{\mathrm{d} z} = e^z =x $$ ...
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1answer
111 views

Hint for solving $ y (y')^2 + (x-y) y' - x = 0$

Need to solve the following ODE: $$ y (y')^2 + (x-y) y' - x = 0$$ I don't really know how to start. Any hints?
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1answer
203 views

Norm bound on exponential matrix with eigenvalue negative real part, proof

If $A$ is $n \times n$ with negative real parts of all eigenvalues, then there exists positive $K,\alpha$ such that $$\|e^{At}\| \leq Ke^{-\alpha t}$$ Furthermore, if an eigenvalue has negative part ...
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2answers
31 views

Solving $\frac{b}{a-b}e^{at}=\frac{x(t)}{a-x(t)}$ for $x(t)$

I`ve been trying to solve the differential equation $x(t)'=x(t)(a-x(t)), x(0)=b, t\in [0, \infty]$. Using the technique of seperation of variables, I get $\frac{b}{a-b}e^{at}=\frac{x(t)}{a-x(t)}$. Now ...
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1answer
84 views

How to integrate $\int \frac{e^x \cos x}{\tan x+\operatorname{sec}x}dx$?

How to integrate: $$\int \frac{e^x \cos x}{\tan x+\operatorname{sec}x}dx$$ I don't really have a clue? Do I need to simplify it first somehow?
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2answers
91 views

What makes a differential equation, linear or non-linear?

Among these differential equations why one is linear while other is non-linear? What is criteria to find out whether a differential equation is linear or non-linear?
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1answer
41 views

Ordinary differential equation . [closed]

The roots of the auxiliary equation for a homogeneous linear differential equation with real constant coefficients that has $ y= 4 + 2x^2 - e ^{-3x}$ as a particular solution are : 1) $ m= 0 , 0 , ...
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0answers
49 views

Proving this is the unique solution to this simple system of diff equations.

So the set of equations are these $\frac{d \omega_x}{dt}+\Omega \omega_y =0$ $\frac{d \omega_y}{dt} - \Omega \omega_x =0$ You can easily differentiate again, get two second order linear diff ...
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0answers
94 views

how to rearrange matrix equation to have unknown in vector form

I am looking for the name/type of following equations: $$\dot{\theta}\dot{J} = \ddot{x} - J\ddot{\theta}$$ here the unknown is $J \in R^{m \times n}$, $x \in R^{m \times 1}$, $\theta \in R^{n \times ...
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1answer
65 views

Show $k$-form/chain identity

Let $\omega$ be a closed $k$-form on $\mathbb{R}^n$ and $c:I^k\rightarrow\mathbb{R}^n$ a $k$-cube on $\mathbb{R}^n$. Let $\mathbb{X}$ be a vector field on $\mathbb{R}^n$ with flow $\Phi_t$. Show that ...
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1answer
233 views

Closed 1-forms implies exact 1-forms

I have two problems, the first one I think I've proved, but I have problems on the second one. Let $\omega$ a closed $1$-form defined on a open $U\subset \mathbb{R^2}$ and let $\gamma:[0,1]\rightarrow ...
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1answer
51 views

Stability of a system of differential equations of the form $\dot x = y, \dot y = g(x)$

Let $g: \mathbb{R} \to \mathbb{R}$ be a locally Lipschitz-continuous function with $g(0) = 0$ and $xg(x) < 0$ for all $x \neq 0$. Consider the differential equation $\dot x = y, \dot y = g(x)$. I ...
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86 views

How do I solve first order non-linear system of PDE: $\partial f^i(x,y)/\partial z = F^i(f^1,f^2,…,f^n)$?

Suppose that I have a system of PDEs of the following form: \begin{eqnarray} \frac{\partial f^i(x,y)}{\partial z} = F^i(f^1,f^2,...,f^n), \qquad i = 1,..,n \end{eqnarray} Where $z = x + iy$, ...
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2answers
523 views

Coupled second-order differential equations

I am trying to solve the following system of coupled ODEs: \begin{align} -x^2 f'' - 3xf' + (1-2a)f - (a+1)x^2g'' + (2-4a)xg' + (4a-2)g &= 0,\\ (a-1)x^2 f'' + (4a+2)xf' + (12-6a)f + 12xg' + ...
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3answers
254 views

Solution of Differential equation

Question: Find solution of differential equation $$ 3e^{4x} \frac{dy}{dx} = -16\frac{x}{y^2} $$ which satisfies the initial condition y(0)=1 Solution: I know that I have to bring it in the general ...
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3answers
333 views

Particular solution of $y'' -3y' + 2y = e^t$

I'm trying to find a particular solution of $$y'' -3y' + 2y = e^t$$ My fundamental set is: $$y_1 = e^{2t}\\y_2 = e^t$$ So I chose $y_p = A t e^t$, which gives me:$$y_p' = Ae^t + Ate^t\\y_p'' = 2Ae^t ...
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0answers
129 views

Steady-state of `degenerate' delayed differential equation

Consider the simple delayed differential equation: $$X'(t) = -a X(t) + a X(t - d)$$ where $d$ and $a$ are positive constants. I'm interested in the possible steady-state (stationary) solutions of this ...
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1answer
44 views

homogeneous first order differential equation

is there a method to solve $$\dfrac{dy}{dx} = f(x,y)$$, where $f(x,y)$ is a homogeneous function. I found some examples like $f(x,y)=(x+y)^2$ where it can be solved after converting it to Ricatti's ...
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1answer
40 views

Existence of the first weak eigenvalue of the Laplacian in a bounded domain

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain and $$H:=W_0^{1,2}(\Omega):=\left\{u\in L^2(\Omega):\nabla u\in L^2(\Omega)\right\}$$ be the Sobolev space, where $\nabla u$ denotes the weak ...
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2answers
47 views

Applying Chain rule to $z = z(u, v) = f(x(u, v), y(u, v))$.

If $z = z(u, v) = f(x(u, v), y(u, v))$ is a differentiable function, where $x = x(u, v) = u^2 − v^2$, $y = y(u, v) = 2uv$, show that $$\frac{∂^2f}{∂x^2} +\frac{∂^2f}{∂y^2} =\frac{1}{4(u^2 + ...
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1answer
77 views

graphing the solution of $y'=x^2-3$

I have a Ordinary Differential Equation(ODE) and I got the solution as ​ Now I want to draw graph? How can I do that? I think: ...
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1answer
66 views

How to find correct trial solution of a higher order differential equation?

I have to find correct trial solution of this equation: $$y''' + 3y'' + 2y' = t + \cos t$$ Attempted work: $$r^3 + 3r^2 + 2r = 0$$ $$r(r^2 + 3r + 2) = 0$$ $$r(r + 1)(r + 2) = 0$$ $$r= 0,-1,-2$$ ...
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1answer
108 views

Show that $\frac{\int_\Omega|\nabla u|^2+\int_\Omega\alpha|u|^2}{\int_\Omega|u|^2}$ attains a minimum in $W_0^{1,2}(\Omega)$

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain $H:=W_0^{1,2}(\Omega)$ be the Sobolev space $|\;\cdot\;|_p$ be the seminorm $$|u|_p^p:=\int_\Omega|\nabla u|^p\;d\lambda^n\;\;\;\text{for ...
5
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3answers
226 views

Given the differential equation, how to solve the y function with x as the independent variable?

$y\frac{dy}{dx} = x(y^4 + 2y^2 + 1)$ $y = 1$ when $x = 4$ I tired to integrate by substitution, but it doesn't seem to work out.
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2answers
317 views

differentiate matrix exponential

I know this: $$\frac{d}{dt}e^{At} = Ae^{At}$$ However, in one lecture, I find the following: $$\frac{d}{dt}e^{A^Tt} = e^{A^Tt}A^T$$ The lecture is as following: How to show the second case, ...
5
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2answers
75 views

LaSalle invariance, Lyapunov stability

Trying to understand the LaSalle invariance principle. Consider the system $x' = y \\ y' = -y-6x-3x^2$ a) Using the test function $V(x,y) = 0.5y^2+3x^2+x^3$, show that the origin is asymptotically ...