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

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clarity in the solution of the following problem

$$(D^2+D)y=x^2+2x+4$$ I found the solution as $$CF=C_{1}+e^{-x}C_{2}$$ and PI=$$\left(\frac{x^3}{3}\right)+4x$$ but the solution from my teacher is PI = $$\left(\frac{x^3}{3}\right)+4x+C3$$ Where ...
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38 views

Can I have variables extreme of integration?

Suppose you have a function $v(t)$ that you want to find. The condition is that it's integral is some fixed quantity. The integral is done between $0$ and $u(t)$, where $u(t)$ is an increasing ...
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20 views

Develop a concept of weak solvability for $-\langle\nabla,A\nabla u\rangle=f$

Let $\Omega\subseteq\mathbb{R}^n$ a domain $f\in L^2(\Omega)$ $A:\Omega\to\mathbb{R}^{n\times n}$ be Borel-measurable with $A(x)$ is symmetric, for all $x\in\Omega$ $\exists c_1,c_2>0$ with ...
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29 views

If $\partial\Omega\in C^{2+\alpha}$ and $-\Delta\Theta=f\text{ in }\Omega$ with $f\in C_0^\infty(\Omega)$, then $\Theta\in C^{2+\alpha}$

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain with $\partial\Omega\in C^{2+\alpha}$ for some $\alpha>0$ $f\in C_0^\infty(\Omega)$ $\Theta\in C^0(\overline{\Omega})\cap C^2(\Omega)$ be the ...
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38 views

What is the differential equation, given a certain solution?

I am a little stuck on this problem. The question asks, Write a first order autonomous differential equation such that $y(t)=\cos(t)$ is a solution. I understand that first order means that it ...
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41 views

Solve x' = -2x + 2y, y' = 2x - 5y

Solve the equations $$x' = -2x + 2y$$ $$y' = 2x - 5y$$ I got $$x=2(c_1) e^{-t}+(c_2) e^{-6t}$$ $$y=(c_1) e^{-t}-2(c_2) e^{-6t}$$
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21 views

how to get this specific matrix differentiation?

Currently, I am reading this article with title "neighborhood components analysis", http://papers.nips.cc/paper/2566-neighbourhood-components-analysis.pdf. Everything went well until I encounter the ...
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46 views

Newton's Law of Cooling Thermometer Taken Back

At 9 a.m., a thermometer reading 70F is taken outside where the temperature is 15F. At 9:05 a.m. the thermometer reading is at 45F. At 9:10 a.m. the thermometer is taken back indoors where the ...
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66 views

Comparing integrals over the unit square

I'm having trouble proving the inequality in the problem below. Any guidance would be greatly appreciated. Let $P:= [0,1]\times [0,1]$ and suppose $\varphi : P \rightarrow \mathbb{C}$ is a smooth ...
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26 views

Is it possible to construct a green function of the Dirichlet problem from the green function of the Cauchy problem?

For the heat equation. Is there a method to obtain the green function of the Dirichlet problem in a rectangular 2D domain from the green function of the Cauchy problem (infinite domain) PDE's?
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43 views

How to motivate those expansions?

I've been reading a paper where the author needs to solve the biharmonic equation on the plane. In truth, the function being saught is a function $v$ such that $v = \nabla \times U$ and $\nabla^4 U = ...
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16 views

Method of Solving for Differential Equations with Trinomials

Solve for $((D^5 + 5D + 4)*(2D^2 + 5D + 2))y = 0$ Answer is $y = c_1*e^{-4x} + c_2*e^{-x} + c_3*e^{-2x} + c_4*e^{\frac{-x}{2}}$ I'm having a hard time looking for a method to solve this; Any hint on ...
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25 views

Solving a 2nd-order ODE via Eigenvectors/values

I am trying to solve $x'' - 8 x'+ 25 x = 0$. I found the general solution as $Ae^{4t}cos(3t) + Be^{4t}sin(3t)$, where $A,B$ are arbitrary constants. However, when I write the differential equation as ...
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14 views

A 2-variable function with parts of its derivative

Let $f$ be a function with partial continuous derivatives, The derivative of $f$ in $ A=(2,2) $ in the direction from $A$ to $B=(2,5)$ is $6$ The derivative of $f$ in $ A=(2,2) $ in the direction ...
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18 views

Scalar product with $f(x,y)$ when $ \dot{X} = f(X)$ has periodic orbit

Let $g(x,y), f(x,y) \in C^1: R^2 \to R^2$ such that $f(x,y) \cdot g(x,y)=0$, $\forall (x,y).$ Prove that if $\dot X = f(X)$ has a periodic orbit, then $g$ have a root Intuitively I can see that the ...
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20 views

Given a 2-variable function, prove that: $xz * z'_x-yz*z'_y = -\frac{1}{2}$

I'm kinda new to this material and was having a hard time solving this exercise, can anyone please show me the correct way of solving this? Let $z(x,y)$ be an equation that satisfies: $ ...
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28 views

this function is known ?: $h_{n+1}(x)=h_n(x)^2+h′_n(x)$

Let $f(x)=x^{1/x}$, so the first derivative of $f(x)$ is $f′(x)=f(x)∗(1−ln(x))/x^2$, and in general, $f^n(x)=f(x)∗h_n(x)$, where $f^n(x)$ is the nth derivative of $f(x)$. I was trying find this ...
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40 views

Time period of oscillations of a point about the function's minimum value?

How am I to go about the following problem? Please do not explicitly solve it. Let $E_0$ be the value of the potential function at the minimum point $\xi$. Find the time period $T_0=\lim_{E\to ...
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27 views

How to turn the integro-differential equation into an ODE

I want to get the numerical solution of the integro-differential equation by Mathematica but failed. Maybe the first step should be turning that into an ODE, is there some method? {0.01+10 (0.01 ...
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9 views

Complex 2 variable differential equation of order 4

Is there any way to obtain all solutions od the following equation: $$ u^{IV}+A\ddot{u}+B\ddot{u}''+i\dot{u}''=0,$$ where $u=u(z,t)$, $z \in \mathbb{C}, t >0$, the dot represents a derivative with ...
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25 views

How to get this numerical solution of a integro-differential equation

Previously, I ask an NDSolve questions(How to solve the differential equation with Duhamel's integral?), but now a more sophisticated one: ...
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10 views

Nonuniform partition - euler method

Consider a nonuniform partition $a=t_0< t_1< \dots < t_{\nu}=b$ and assume that if $h_n=t^{n+1}-t^n, 0 \leq n \leq N-1 $ is the changeable step, then $\min_{n} h_n > \lambda \max_{n} h_n, ...
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6 views

Simple inverse laplace using partial frac not so simple?

When evaluating the step response of a circuit, the resulting Laplace representation is: $\frac{I_{pd}}{s^2 C1 R1}$ If I look this up on a table of Laplace Transforms, this results in ...
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12 views

Absolute stability Euler method

I am looking at the following exercise. We suppose that the explicit Euler method is applied at the differential equation of second order $\left\{\begin{matrix} x''(t)+(\lambda+1)x'(t)+ \lambda ...
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35 views

Differential equation - fractions, circular answer?

Hi this might seem like a really stupid question but then hopefully someone can asnswer it quite easily :) I have function $P{_t}$$=(E{_t}$ $(P{_t}{_+}{_1}+$ $δ{_t}{_+}{_1}$$ )-γΩx$${^*})/$$(1+rf+ψ_t ...
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27 views

Integro-Differential Equations

I was looking for elementary examples of integro-differential equations, and I wondered would the following equation, $\frac{d^{2}x}{dt^{2}} = \frac{GM}{x^{2}}$ qualify as a valid example? Because it ...
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14 views

Rosenbrock Method Implementation (order 2,3)

for the solution of a (stiff) inhomogeneous 4th-order-ODE I use the Rosenbrock-method which is implemented in matlab (ode23s). I now have the problem that I have to move to implement such a method by ...
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24 views

Using Orthogonal Collocation to solve Coupled Ordinary Differential Equations

I am trying to solve six first order coupled ODE's, two of these are associated with a heat balance of a catalyst pellet, and four are mass balances. I have been trying to solve these equations using ...
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30 views

Solve first order BVP on MATLAB

I try to solve this equation on MATLAB $$y^\prime(x)=x+y^2(x)\quad\forall x\in(0,0.9),\quad y(0)=1, y(0.9)=32.725$$ I write two function on matlab, on myODE.m file, ...
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19 views

General solution to this ODE

Was reading through my notes on Fourier transform but couldn't figure how to find the general solution to $$\frac{\delta U(\omega ,t)}{\delta t} = -k\omega ^{2} U(\omega ,t)$$. I must definitely have ...
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23 views

stability of $ \dot x = (x-1)(y-2), ~~~\dot y=(x-3)(y-2)$

Question: I want to determine the point of equilibrium and the stability (asymptotically stable, stable, or intable) $$ \dot x = (x-1)(y-2), ~~~\dot y=(x-3)(y-2)$$ Attempted solution: So it has to ...
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32 views

Gronwall's Lemma with an additive term

Suppose that $V(t)$ is a real function of $t\in \mathbb{R}$, differentiable on $t>t_0$, and satisfies $$\dot{V}(t) \le -\alpha V(t) + \beta \sqrt{V(t)}\quad \forall t>t_0,$$ where ...
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32 views

How can I show that $x'(t) = - (1-x(t))'$ from $x(t)$'s functional form?

I have a differential equation modelling the concentration of vaccinators in a population. Here are a few assumptions. Assume we can write the concentration of vaccinators as $x$, and that we can ...
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17 views

How to show that these fractions converge?

Considering the following differential equations, how can I show both S(t) and R(t) converge as t goes to infinity? \begin{equation} \frac{dS(t)}{dt}=-aS(t)I(t) \end{equation} \begin{equation} ...
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19 views

Decoupling system of three second order ODE

I have the following set of equations: $\begin{alignat}{3} &f'' + Af' - (~~X~+~B~~~)f ~~~~- Ug &= 0\\ &h'' + Ah' - (~~Y~+~B~~~)h ~~~~- Vg &= 0\\ &g'' + Ag' - (X+Y+B)g - Uf ~+ Vh ...
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84 views

$\frac{d^2 \theta}{dt^2}=9*\frac{d^2 \theta}{dx^2}$ use method of separation of variables to find final solution $\theta(x,t)=$?

Boundary conditions: $\theta_x (0,t)=\theta_x(4\pi,t)=0$ $t>0$ Initial conditions: $\theta(x,0)=3\cos 2x$, $\theta_t(x,0)=1+6 \cos 2x$, $0<x<4\pi$ Use result: $$X''-\lambda*X=0$$ ...
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26 views

reducing this Bessel differential equation using euler's ode

$$\frac{d^{2}y}{d^{2}z} + y ( 1 + \frac{1}{4z^{2}} - \frac{m^{2}}{z^{2}}) = 0$$ I have the above DE which I managed to transform into using a scaling transformation in response to a prior question ...
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55 views

How can I solve this variable-coefficient ODE system?

I originally have a linear, homogeneous, second-order variable coefficient ODE system of this form: $X''(x) = A(x)X(x)$, where $X(x) = $\begin{bmatrix} f(x) \\ g(x) \\ \end{bmatrix} ...
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39 views

ODE: differential notation

I was taught in Calculus classes that the symbol $dx$ may stand for various things, for example, when we are integrating a function $dx$ tells us that we are integrating with respect to the variable ...
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17 views

Solve by method of finding the integrating factor: $dy/dx = 5y$

I found that the integrating factor is $e^C/e^{5x}$ and after integrating these were my equations: $$ \int Mdx = ye^C/e^{5x} + f(y) $$ $$ \int Ndy = ye^C/e^{5x} + g(x) $$ Comparing the two ...
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Differential equations, theoretical question about lowering the power of a basic differential equation.

In the text book it says we can solve:(The area of existence and uniquesness of the equation is $G= R \times R^n$)$$x^{(n)}=f(t), f \in C(R) \tag{1}$$ the following method: Integrating (1) $n$ times ...
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96 views

Total differential proof , need help understanding. Integration factor.

Now we're trying to find a solution for: $$ \mu(t,x):\qquad(*) \frac{\partial \mu}{\partial x}P- \frac{\partial \mu}{\partial t}Q + \mu\left(\frac{\partial P}{\partial t} - \frac{\partial Q}{\partial ...
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linear term of $\sqrt{ 1 - A \exp^{2iwt} - B exp^{2iwt}}$ does it make nonsense to analysis of dynamics?

I have the following problem: I am building a Lagrange Euler equation near the position where some vector component equals to 1, while others no. I do the following substitutions: $$\vec x = ...
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17 views

How to check if a solution is symptotically stable equilibrium.

I have the differential equation : $$4x^2y''+y=\sqrt{x}$$ $$x>0, \,\,\ y(1)=0 ,\,\,\, y^\prime(1)=0$$ after solving this problem with 2 different ways, I got the solution: $y=\frac{1}{8} ...
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20 views

Variation of parameters, in one case we assume something in one we don't, but we end up with the same result?

You have a second order differential equation $y''+P(t)y'+Q(t)y=f(t)$. And you have two independent solutions to the homogeneous system, call them $y_{h_1}, y_{h_2}$. In order to find a particular ...
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61 views

Exact solution to the given system of ODE 1

I'm trying to better understand basic neuroscience systems but I have almost no background in differential equations. Here's the standard leaky integrate-and-fire neuron with conductance based ...
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17 views

If a Point Admits an Integral Curve on an Interval then a Neighborhood Does too On the Same Interval

$\newcommand{\R}{\mathbf R}$ Let $V:U\to \R^n$ be a (continuous) vector field on an open set $U$ of $\R^n$. Suppose we have a point $\mathbf p\in U$ and an open interval $I\subseteq \R$ such that ...
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22 views

The Arbitrary Lagrangian Eulerian (ALE) description

Considering that in an ALE framework, the partial derivative relation $\frac{\partial}{\partial t}=\left.\frac{\partial}{\partial t}\right|_{x}+\underline w.\nabla$ where $w$ is the mesh velocity, ...
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16 views

Construction of function with properties and specific values

I would think something of this nature would be solved using differential equations such as is done with recursion relationships. So let's say we have a continuous SMOOTH function, increasing so that ...
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36 views

A question in partial differential equation.

Suppose a.b is bounded in $L^2(0,T;H^{-1}(\Omega))$; $a\geq \alpha>0$ almost everywhere in $\Omega\times(0,T)$ and $a\in L^\infty(\Omega\times(0,T))$. Is $b\in L^2(0,T;H^{-1}(\Omega))$?