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

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Charge distribution in a conducting sphere

This is a math problem that appeared in a physics one: "Evaluate the density of electrical charge in the surface of a hollow sphere such that the electric field is constant (in direction and absolute ...
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An differential problem is general

Does there exist a differential equation $g(f^{n},f^{n−1},...,f,x)=0$, such that the corresponding equation $g(x_n,x_{n−1},...,x_1,x_0)=0$ has solutions in $C^{n+1}$, but that itself has no solution?
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Several questions regarding existence of a unique solution for nth order differential equations

I am trying to understand this theorem in my Differential Equations book: Let $a_n(x), a_{n-1}(x),...,a_1(x),a_0(x)$ and $g(x)$ be continuous on an interval $I$ and let $a_n(x) \ne 0$ for every x in ...
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Approximation error of Runge Kutta 2nd order

I am simulating a dynamical system, using Euler's method to solve ODEs which resulted in huge error given the step. Then I shifted it to rk4 but it was pretty slow, now I use rk2 considering the ...
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15 views

Prove that the solution of the differential equation exists for $0\le x\le min(a,{b\over a^2+b^2})$

Consider the initial value problem $y'=x^2+y^2, y(0)=0$ and let $R$ the rectangle $0\le x\le a$, $-b\le y \le b$. Prove that: The solution $y(x)$ exists for $0\le x\le min(a,{b\over a^2+b^2})$ I ...
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Ornstein-Uhlenbeck -> AR(1)

Any suggenstions to where I can read a rigorous proof of how the descrete time version of an Ornstein Uhlenbeck process can be considered as an AR(1) process?
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Using Laplace transform to solve the ODE $DP/dt = P + k(H(t-T)-1)$

A population of fish that's growing is harvest at a rate of $k$, from $t = 0$ to $t = T$ and follows the DE: $$DP/dt = P + k(H(t-T)-1), \qquad P>0$$ How do I solve the IVP with $P(0) = P_0> ...
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Different types of critical point $(S.L Ross) $ Article 13.3

I am studying critical points in differential equation: Consider the system of differential equations $$ \frac{dx}{dt} = ax + by$$ $$\frac{dy}{dt} = cx + dy$$ where $a,b,c,d$ are real constants ...
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Logistic Model involving a population already at max capacity

QUESTION: Suppose $P(t)$ is the mass of a pollutant in kilograms in a lake at time $t$ in weeks. Further suppose that fresh water flows into the lake at $5 × 10^8$ L per week and the same amount of ...
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Analytical solution to nonlinear ode

I solved this equation that I attached numerically in matlab by the Newton Raphson method. Now I want to solve it analytically in matlab or even in Maple if it is possible. Would you please help me ...
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Q: Second Order Differentiation

I have this differential equation question: $$ y''-3y'+2y=\frac{e^x}{1+e^x} $$ I can get the general solution for the homogeneous equation. However, I can't get the particular solution. It's the ...
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Differential equation involving composition

I've been studying Euler's method to approximate a solution to a differential equation in an algorithm class. I faced a weird differential equation in a mathematics exercise, and I wanted to know if ...
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19 views

Differential Equation for Algorithm Time

I'm working on algorithm analysis and time complexity. I've got a homework assignment to calculate a function f(n) at time t and I want to figure out how to write it as a differential equation. ...
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26 views

Which model technique to use for differential equation

We have a diferential equation: $$u'(t) - u(t) = y'(t) + y(t)$$ Which method do we use to create a model in simulink for this diferential equation, as we cannot use Indirect method, becouse ...
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Why the sense of orientation in the graphic representation of a 2-form does not cancel each other?

In this article, it is said that the graphic representation of the 2-form ${\bf F}=B_z{\bf d}y\wedge{\bf d}z$ are tube with a sense of a circular orientation. How does this circular orientation show ...
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63 views

Where is this equation coming from?

I'm given a problem and I've finished the problem, but I was just wondering where we are getting an equation from. The problem is pretty simple and involves plugging values into the equation. When I ...
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51 views

Show that z satisfies the initial value problem

Consider the cohort of individuals born in a given year(t=0), and let n(t) be the number of these individuals surviving t years later. Let x(t) be the number of members of this cohort who have not had ...
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The implication of the Uniqueness of Cauchy problem and maximum existence interval of 2 distinct solutions

Hello I wish to quickly refresh what the uniqueness of Cauchy problem is telling us, well roughly speaking at least. So, suppose we have two solution, $u_1$, $u_2$ with initial data $u_{1,0}, ...
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Lyapunov exponents of a linear upper block triangular system

I seem to be stuck at formally showing something that intuitively seems to be true. I have a linear non-autonomous system of the form $$ \dot{x} = A(t)x $$ where $A(t):\mathbb{R}\to ...
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move forward with quadratic rates of change relationship

I am examining the relationship between two time-sensitive variables, $f_1(t)$ and $f_2(t)$. If I plot $df_1/dt$ agains $df_2/dt$, I find a parabolic line, $df_2/dt = k - (df_1/dt)^2$ where k is a ...
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Transfer function: steady-state solution of equation

Place the transfer function in the form $$H(i\omega) = \frac {1}{R}e^{-i\phi}$$ and use this result to find the steady-state solution of the equation$$x'' + x'+4x = 3*cos(2t)$$ I don't really ...
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How to solve boundary condition?

$$\begin{align}\frac{d^4 y}{dx^4}&-y=0\\y(0)&=1,\; y'(0)=1\\y(1)&=1,\;y'(1)=1\end{align}$$ I get $$y(x)=Ae^x+Be^{-x}+C\cos(x)+D\sin(x)$$ After applying conditions ...
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Differential equation drug administration

A drug is administered intravenously at a constant rate of r mg/hour and is excreted at a rate proportional to the quantity present, with constant of proportionality k>0. (a) Set up a differential ...
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Probablistic method (Erdös) Differentialequations (Peano)

I have run into this: http://en.m.wikipedia.org/wiki/Probabilistic_method Now I am very curious to see an application. I do know that this is rather specific but i thought, there might be situations, ...
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small initial data

Consider a simple ODE problem of Cauchy type, after local existence and uniqueness is established, one seeks for global existence, now, my question is, Why does one need "small initial data" to ...
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Find coefficients for harmonic oscillator ODE

Related to Small mismatch in theoretical vs MMA solutions of a damped harmonic oscillator differential equation, given the ODE $$ \ddot y + 2\xi \omega \dot y + \omega^2 y = fo \cos( \omega_{dr}x), ...
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How do you take divergence of this field?

I forgot how to do divergences 3 years ago, this one is very confusing. We used to take them with respect to $x,y,z$ but this one doesn't have them. Help me with this question Prove that $∇.E = 0$ ...
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34 views

Solution for trading algorithm equation

Do anybody know what does this equation means.? I have seen this Kf coming above. But not downside.
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29 views

Eigenvector of canonical form matrix

So if you have the canonical matrix: $$ \begin{pmatrix} \lambda &1\\ 0 & \lambda \end{pmatrix} $$ the eigenvector is $(1,0)$. I've found this from multiple sources. But where does that come ...
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Problem Analysis - Answer but no procedure - Finding Isogonal Trajectories.

I stumbled with this problem in a notebook that has been bothering for the whole day(actually 4)...The answer is written but there's no explanation nor a steb-by-step procedure or anything. If you ...
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model position of bicycle wheels

Given the position $f:[0,T] \to \mathbb R^2$ of the front wheel of a bycicle and the distance between both wheels $L > 0$ I want to find the position of the backwheel $r:[0,T]\to \mathbb R^2$ for ...
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Using Frobenius to show two solutions

Consider the ODE $$y''+\frac{y'}{x}+\left(-1-\frac{1}{4x^2}\right)y$$ Use the Frobenius method to find a power series solution near $x=0$ corresponding to the smallest root of the indical equation, ...
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Population growth equation

How can I answer this problem using the equation $P(t) = P(0)e^{rt}$? Not looking for the math to be done for me, I'm just a little confused with what should be assigned to what variable. Biologists ...
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Integrating factor (exact equations)

$$(3x + (6/y)) + (x^2/y+ 3y/x)dy/dx=0$$ $$M_y=-6/y^2 $$ $$N_x= 2x/y - 3y/x^2$$ how do I go about finding the integrating factor for this equation? thanks for tips/solutions
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Solving a system of coupled 'differential' equations

Over the course of working on a project I have arrived at the situation where I have the following two (simplified) equations where each has a dependency on the other $x = \frac{1}{y}$ ...
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$dy/dx=e^{-x^2} , y(2)=6$

hey so I'm trying to solve this equation using separable method ( an example from my book). It says it isn't an elementary function ( I don't really understand that, but a function that can't be ...
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Laplace Transform To Solve IVP

I need to use the Convolution theorem to solve $y'' +4y' +4y = g(t)$ with initial conditions $y(0)=2, y'(0)=-3$. This is what I have but it differs from the answer in the text so I'm wondering where I ...
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Problem with an Ordinary Differential Equation Problem

So I have to solve $2t^2y''+(y')^3=2ty'$, I start by making $v=y'$, so then I have $2t^2v'+v^3=2tv$, divide whole equation by $2t^2$, so $v'+v^3/(2t^2)=v/t$, where this is Bernnoulli, so I let ...
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The method of undetermined coefficients

What's the proof behind the method of undetermined coefficients that's used in solving second order non-homogeneous differential equation with constant coefficients?
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Solving a Gompertz equation

(a)Solve the Gompertz equation $$dy/dt = r y \ln(K/y)$$,subject to the initial condition $y(0) = y_{0}$. (b) Data given $[r = 0.71 , K = 80.5 × 10^6 , y_{0}/K = 0.25]$, use the Gompertz model to ...
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Dependent and independent variables of a differential equation

$$\theta'' + \delta\theta'+\sin\theta = F\cos(\omega t)$$ I am trying to write it as a first order equation, and state the dependent, independent and parameters in the ODE. I have written it as: ...
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Solving System of 2 simple odes

Before I used euler method, it didn't seem a good approximation. Thus, I have shifted to ranga kutta method. I am just trying to solve two simple odes (dyanmical model) using Runge-Kutta method: ...
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Let $(I_\eta, y_\eta)$ be maximal with $y_\eta(1) = \eta$ (IVP). Show for $0 < \eta < 1$ we have $y_\eta(t) < t^{\frac 4 3}$, $t \in I_\eta$.

Consider the differential equation $y' = X(t,y)$ with $X(t,y) = \frac 1 3 y^{\frac 1 4} + t^{\frac 1 3}$, defined on $\mathcal D_X = (0,\infty) \times (0,\infty)$. For $\eta > 0$ let $(I_\eta, ...
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Asymptotic solutions to generalized Airy equation

I am interested in asymtotic solutions, for $x \gg 0$ and $x \ll 0$ of the following differential equation: $\frac{d^ny}{dx^n} + yx = 0$ Here $n$ is an integer $\ge 2$. For the particular case of ...
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Determine the equilibrium temp distribution for a 1D rod with the following sources and boundaries.

Q=0 du/dx(0) =0. u(L)=T So ,my attempt is that u(x) = Ax + B, so du/dx = A implies A=0 and so u(L) = 0 + B = T so the solution becomes u(x) = T. But I have a feeling it's not right or I'm ...
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first-order differential equation problem

Given that $y=\sin(x)$ is an expicit function of the first-order differential equation $\frac{dy}{dx}=\sqrt{1-y^2}$. Find an interval I of definition, the solution interval. So I got to the point ...
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Comparison theorem for parabolic partial differential equations

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain $J\subseteq\mathbb{R}$ be an intervall $T\in(0,\infty)$ and $f\in C^0\left(\overline{\Omega}\times[0,T]\times J\right)$ be locally Lipschitz ...
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Second order linear ODE arising from Euclidean heat kernel

When solving for the Euclidean heat kernel $H(t,x,y) \in C^{\infty}((0,\infty) \times \mathbb{R}^n \times \mathbb{R}^n)$, one way to proceed is to look for a solution in the form $H(t,x,y) = ...
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Is it possible to show the uniqueness of formula for solution?

The motivation to this question can be found in: Show that any sequence $(u_{n})$ must tends to infinity as $n→∞$ My question is: Is it possible to show the uniqueness of the formula for the ...
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Substitution in a system of ordinary differential equations when terms of the same order derivative for different variables occur in the same equation

Let's say I have a differential equation such as: y'' - 2ty' + y = 0, y(0) = 2.1, y'(0) = 1.0 I can solve this (among other ways) by substitution and conversion ...