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

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Find Population from Model

First time I asked this it didn't get entered correctly so got wrong answers, which helped explain a different question I had. $$\frac{dp}{dt} = \frac{2p(7-p)}{700}$$ Given $p(0) = 3$ and $p \gt ...
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0answers
6 views

Jump Diffusion Infinitesimal generator

I have this difussion process $dX(t)=\mu X(t)dt+\sigma X(t)dW(t)+u X(t) dN(t),\qquad X(0)=x > 0$ where $W(t)$ is a Brownian Motion and $N(t)$ is a Poisson process. And I need to know the ...
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0answers
10 views

Maximal interval of solutions existence: $x'(t)=-x(t)+\sin x(t)+t^3$

$$x'(t)=-x(t)+ \sin x(t)+t^3$$ I consider the function: $$ f(t,x)=-x+\sin x + t^3 $$ $$\frac{\partial f}{\partial x}=\cos x-1$$ I see that: $$\left| \frac{\partial f}{\partial x} \right| < 2$$ ...
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31 views

Solution of $x'=Ax$ is not what it is supposed to be

Consider a system of ODEs $x'=\begin{bmatrix} 0 &1 \\ -1&0 \end{bmatrix}x$. Wolfram Alpha says that the solution is $x(t) = \begin{bmatrix} \cos t &\sin t \\ -\sin t& \cos t ...
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0answers
17 views

Solving differential equation containing function compositions

Is it possible to solve differential equations with composite functions, e.g. $f^{(n)}\circ (x-1)$ and $f^{(n)}\circ(x)$. I'm particularly interested in $$2f'(x-1)+(x-1)f''(x-1)-f''(x)=g(x).$$ Do ...
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1answer
10 views

Definition of trajectory

I am writing something that involves comparing the solutions of many different differential equations, and I need precise definitions of the terms trajectory and solution curve. Given a dynamical ...
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2answers
19 views

Canonical form and fundamental solution of pdf

Can someone help with these two PDE problems? Thank you. Reduce to Canonical form and find the fundamental solution if possible. $$y^2u_{xx} + x^2u_{yy} = 0.$$ What type of transformation should I ...
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2answers
36 views

Seperation of variables is not rigorous enough.

So I have started ODE's and have been told that solving initial value problems by using separation of variables(like I had been taught) is nothing but a leap of faith. Now we must use something along ...
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1answer
18 views

Ex ODE: $y'=4t \sqrt y- \lambda(y-(1+t^2)^2)$

How to solve the following equation? $y'=4t \sqrt y- \lambda(y-(1+t^2)^2)$ $y(0)=a$ Show those cases where a numerical method will solve this equation exactly. $(a,\lambda) \in {\Re}^2$
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2answers
26 views

Solution Set of System of ODE.

I am trying to find the solution of the system $$\begin{bmatrix}x_1\\x_2\end{bmatrix}'= \begin{bmatrix}1&3\\3&1\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}$$. I am given that ...
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25 views

How to calculate inverse laplace of $e^{a\sqrt s}$?

I was using Laplace to find solutions for $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}$$ with boundary conditions $$u(0,t)=1 \\ u(1,t)=1 \\ u(x,0)=1+ \sin \pi x$$ I used ...
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Stability of gradient flow $x'(t) = -\nabla E(x)$, $E(x)$ is convex function. [on hold]

Let $E: R^d \to R$ be convex and continously differentiable. The IVP $x'(t) = -\nabla E(x), x(0) = x_0, t>0$ is called the gradient flow. Show that gradient flow has following properties a) ...
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2answers
23 views

Explicit Euler Method's Order

I am seeing everywhere that the order of the Explicit Euler Method is 2 but I can't prove it on my own. The textbook that I found the method says that the proof is very easy so it is up to the reader ...
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1answer
31 views

How to use the 3rd and 4th boundary conditions in this?

I was solving $$ \frac{\partial^2 u}{\partial t^2}=\frac{\partial^2 u}{\partial x^2}$$ All the boundary conditions are as follows:- $$u(0,t)=0 \\ u(\pi ,t)=0 \\ u(x,0)=\sin x \\ u_t(x,0)=x^2$$ ...
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Eigenvalue problem for $−\psi''(x) − (ix)^ N \psi(x) = E\psi(x)$ in complex plane

To find the eigenvalue $E$ in the complex plane of $x$ for one dimensional Schrodinger equation $$ −\psi''(x) − (ix)^ N \psi(x) = E\psi(x). $$ where $N$ can be any real number, the boundary condition ...
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4answers
63 views

Why cannot $A\sin\alpha x +B\cos \alpha x$ be zero?

I was going through solving wave equations using fourier and I came across a note saying $A\sin\alpha x +B\cos \alpha x \neq 0$ I believe this applies to $\alpha ,A,B\neq 0$ I was solving $$ ...
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2answers
16 views

If $g(t)=d$ (constant), show that all solution of $L[y] \to \frac{d}{c}$ when $t \to \infty$

Let the differential equation $L[y] = a y'' + by' + cy = g(t)$, where $a$, $b$ and $c$ are strictly positive numbers. If $g(t)=d$ (constant), show that all solution of $L[y] \to \frac{d}{c}$ when $t ...
2
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1answer
60 views

Doomsday Prediction

I have a calculus problem I can't seem to figure out. Any help would be appreciated! Doomsday prediction. In $1960$, three electrical engineers at the University of Illinois published a paper in ...
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1answer
15 views

Show that $Y_1[t]- Y_2[t] \to 0$ as long as $t \to \infty$ - Differential equations

Let the differential equation $L[y] = a y'' + by' + cy = g(t)$, where $a$, $b$ and $c$ are strictly positive numbers. If $Y_1(t)$ and $Y_2(t)$ are solutions at the $L[y]$ equation, show that $Y_1[t]- ...
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1answer
18 views

Turing criteria for Sel'kvo glycolysis model

I have the Sel'kov reaction diffusion model for glycolysis as follows: \begin{eqnarray} u_t=D_uu_{xx}-u+av+u^2v\\ v_t=D_vv_{xx}+b-av-u^2v \end{eqnarray} How can I obtain the values for $D_u$ and ...
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Searching for second singular solution of a differential equation

I was given the following equation $\frac{dy}{dx}=\sqrt{y-4}$ and was asked to solve it and find a family of solutions. I've solved the equation so that $y=\frac{x^2+16}{4}+c$ or ...
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1answer
21 views

Separable solution to a nonlinear parabolic PDE

I seek a separable solution to the nonlinear parabolic partial differential equation, $\frac{\partial u}{\partial t} = u \frac{\partial u}{\partial x^2} + u^2.$ The physics of the problem allow ...
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1answer
34 views

Evolution semigroups for differential equations

I would like to ask whether "evolution semigroups" are really useful (to discover something that can't be discovered in some other way?). There is a huge machinery to deal with them, but from my point ...
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1answer
15 views

Finite Difference Equation with Constant Co-efficient

I trying to find tutorials on the topic (Finite Difference Equation with Constant Co-efficient) but I can't get exactly what I want. The said Difference Equation has a ...
2
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3answers
31 views

Differential equations (second edition) - William E. Boyce & Richard C. Diprima (#$31$, page $142$)

In many physical problems, non-homogeneous term may differ from one time interval to another. By example, determine the solution $y = \phi(t)$ at $$y'' + y = \begin{cases} t ...
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1answer
20 views

Stable eigenspace of $x'=Ax$

Given the system $\bf{x'}=Ax$, where $\bf{A}$$=\begin{bmatrix} -2 &0 &0 \\ 2& 1 & 0\\ 0 &0 &1 \end{bmatrix}$, the solution is $x(t) = \begin{bmatrix} e^{-2t} & 0 ...
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1answer
19 views

How to find isoclines of the following system?

If I had a system of equations such as: $$\dot{x}=(1+2x+y)x$$ $$\dot{y}=(4+6x+2y)y$$ How would I find the horizontal and vertical isoclines of such a system? I know ...
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22 views

$\bf{x'}=Ax$ with eigenvalues of multiplicity greater than $1$

Given the system $\bf{x'}=Ax$, where $\bf{A}$$=\begin{bmatrix} -2 &0 &0 \\ 2& 1 & 0\\ 0 &0 &1 \end{bmatrix}$, if I solve it by first finding matrix $\bf{P}$ and then ...
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2answers
29 views

Solving this differential equation via dividing by $t^n$

I have this differential equation that I need to solve: $y'=\frac{(x^2-y^2)}{3xy}$ My teacher gave a hint to divide everything on the right hand side by $t^n$, but I do not see how this is helpful. ...
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1answer
33 views

Finding Fourier series constant and integral

I have been studying Griffith's Intro to Electrodynamics. I am studying differential equations and Fourier series. I am studying the problem discussed here: Why is this allowed? ("Fourier's ...
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1answer
21 views

Find the population from a model

$$\frac{dp}{dt} = \frac{2p(7-p)}{700}$$ Given $p(0) = 3$ and $p \gt 0$, find $p(57)$. I'm able to separate the variables and get $$\frac{700}{(2p)(7-p)} = dt$$ I then separate to different bases. ...
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1answer
42 views

$x'=\cos^5(x) +1$ has unique solution defined for all $t\in \mathbb{R}$

I would appreciate if someone could please give me a hint on how to do this problem. Or where to see some examples. Unfortunately, the sources that I have do not seem to actually explain it and show ...
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0answers
10 views

Galerkin method in Sobolev space

I've got this problem to solve: Using Galerkin method, prove that there exists a weak solution of this differential equation: $$-\Delta u = a(x) \circ \triangledown u - u_t +f(x)$$ on $\Omega$ $$u ...
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1answer
25 views

Mixing problems

A tank contains $70$ kg of salt and $1000$ L of water. A solution of a concentration $0.035$ kg of salt/liter enters a tank at the rate $5$ L/min. The solution is mixed and drains from the tank at the ...
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1answer
14 views

Question about a solution of a partial differential equation by separation of variables

I'm trying to understand this text: http://www.ekayasolutions.com/UCDMath/HeatCondSphere.pdf But I'm having problem with this part: Whe have to solve: \begin{equation} \dfrac{\partial ...
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1answer
28 views

ODE problem with a single function but two arguments

I have been trying to solve the following ODE with no success: $$ \frac{df(x)}{dx} = -x f(x) + 4xf(2x)$$ I even tried using Maple but it seems to only accept ODE's that are function of the same ...
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1answer
16 views

Parameters for existence of Solutions for Inhomogeneous BVP Equation.

I've been studying the Fredholm Alternative recently and have become stuck on a couple of questions. What values of $A$ and $B$ will give a solution for the following BVPs? I know that $Q_1$ is ...
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2answers
53 views

Differential equation $y' + yx - 2 = 0$

I'd like to see different methods to solve the differential equation $$y' + yx - 2 = 0.$$ Thank you in advance. Update Since the question has been put on hold as if it misses context I think it best ...
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1answer
19 views

Determine whether $f(x,y) = |y|^{\alpha}$ satisfies the Lipchitz condition.

Where $\alpha \in [0,1),y \in [-1,1]$ It seems to be satisfied, if I choose $\alpha = 1/2,y = 1/2,z = 1/3$ But I can't think how to begin $$||z|^\alpha-|y|^\alpha| \leq \text{???}$$ Could I have a ...
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3answers
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Find the general solution of the ODE $xy′′ − y′ + 4x^3y = 0$ [on hold]

Can someone help me figure out this ODE, its driving me crazy. I dont need a full solution beacuse that would take hours but maybe just the final answer? Find the general solution of the ODE $xy′′ − ...
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2answers
79 views

Solving $\ddot y=\frac{1}{2}e^{-y}$

As the title states I am trying to solve $\ddot y=\frac{1}{2}e^{-y}$ with initial conditions $\dot y(t=0)=0$ and $\ddot y(t=0)=i$, I'm aware of the trick to evaluate this differential but I'm having ...
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2answers
54 views

How to solve the Sturm Liouville problem $y''-3y'+λy=0$, $y'(0)=0$, $y'(\pi)=0$?

Hi everybody I need to find the values of λ>0 and λ<0 to find the own values, I knowk that in the case λ=0 I obtain the trivial solution, but in the other cases I don`t know what to do, any help ...
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1answer
21 views

Find $c, M > 0$ such that $\lvert e^{tA}x_0\lvert \le Me^{ct}\lvert x_0\lvert$

In a system of differential equations $x'=Ax$, where $A$ is a constant matrix, and the equation is a sink (all eigenvalues of $A$ have negative real parts), I need to find constants $c,M>0$ such ...
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0answers
22 views

Difference Equations and Discrete Integral and Derivative

So I'm trying to learn difference equations, and the book that I'm using defines the following: The discrete derivative of a function $a_n$ of the integers is defined as: $$ D a_n = a_{n+1} - a_n $$ ...
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14 views

Undetermined coefficients when RHS is a polynomial

When applying undetermined coefficients to a non-homogeneous linear differential equation, we let $y_p=A_0+A_1x+\cdots A_n x^n$ when a degree $n$ polynomial is on the RHS. Doesn't this potentially ...
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1answer
36 views

Solve $(2x - 1)y'' - 4xy' + 4y = 0$

I am self-studying differential equations using MIT's publicly available materials. One problem set question asks us to first verify that $x$ is a solution to $(2x - 1)y'' - 4xy' + 4y = 0$ (with $2x ...
3
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1answer
41 views

Differential equation where Picard-Lindelöf can not be applied

My question is the following : Let $f:\mathbb{R}\to\mathbb{R}$ be continuous function and let $u:[a,b]\to\mathbb{R}$ be a $C^1$ function such that $$\forall t\in[a,b],u'(t)=f(u(t))\text{ and ...
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2answers
30 views

How to solve the differential equation $y' + \sec(x)*y = \tan(x)$

I am really struggling to solve the differential equation: $y' + \sec(x)y = \tan(x)$. If someone could point me in the right direction or give me a step by step plan it would be much appreciated! So ...
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2answers
33 views

Linear Stability Analysis of an ODE

Question: Find the equilibrium points of the following ODE and then use Linear Stability analysis to determine the stability. $$\frac{dy}{dt}= cy - \frac{y}{1+y^2}$$ My Attempt: I have worked out ...
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1answer
19 views

Confusion regarding the Cauchy-Euler equation

In my textbook it says that: An equation of the form $$a_2 x^2 \frac{\mathrm{d}^2y}{\mathrm{d}x^2}+a_1 x \frac{\mathrm{d}y}{\mathrm{d}x}+a_0 y=f(x)$$ is called ...