Tagged Questions

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

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2
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0answers
5 views

Isolated asymptotically stable critical points of autonomous differential equation

For an equation of the form $\dot{x}=f(x)$ I'm asked to prove that is not possible for a point critical point $a$ to be asymptotically stable and isolated at the same time. Is this statement wrong? ...
-4
votes
0answers
15 views

Boundary Value Problems for the Heat Equation

This tasks are from the book Jeffery Cooper, Introduction to Partial Differential Equations with MATLAB.
3
votes
1answer
17 views

How do you solve a 2nd order differential equation of the form $v = v' - v'' +C^t +D^{t+E}$

I've been working on an economic simulator for a game I've been making and in order to simulate the velocity of money, I created the differential equation of the form $v = v' -v'' + C^t + D^{t+E}$. ...
4
votes
0answers
9 views

Fourier Transform of Newton's Law of Cooling

I am attempting to solve Newton's Law of Cooling differential equation with Fourier Transforms for a high school math report. Can Fourier Transforms be used to solve first-order ODEs? The equation is: ...
4
votes
1answer
31 views

What are the equations modelling a vertical spring system with two masses?

Modeling a vertical spring system with one mass is a pretty common problem. I looked around online and found some horizontal spring systems with two masses, but no examples of a vertical one. I'm ...
0
votes
0answers
5 views

Characteristics metod and unsolvable equations

I am trying to solve: $u_x^2 + u_y^2=u$ with boundary conditions: $u(x,0)=x^2$. Unfortunately it leads to equations that makes no sence (sum of squares is 0 and all constants are 0). I would be ...
0
votes
0answers
17 views

Systems of First Order Linear Equations - Differential Equations

Consider the vectors $x^{(1)}(t) = (t,1)$ and $x^{(2)}(t) = (t^2, 2t)$ I computed the Wronskian which is t^2. But I was wondering how to solve the following questions: 1) In what intervals are ...
0
votes
0answers
12 views

I am required to solve the boundary value problem $y'' = 4x^2y' + 2xy,\space y(1) = 4,\space y(2) = 2$ using the midpoint method.

I am required to solve the boundary value problem $$y'' = 4x^2y' + 2xy,\space y(1) = 4,\space y(2) = 2$$ using the midpoint method. In order to get two first order equations I have set $u_1=y\space ...
0
votes
2answers
20 views

Help with Runge-Kutta method for solving systems of differential equations

I am currently doing an investigation with SIR model for predicting the progress of an infectious disease. However, I am not very much familiar with systems of differential equations,so I would need ...
2
votes
3answers
35 views

Solving $y'(x) = \frac{y(x)}{3x-y^2(x)}$?

Solving $y'(x) = \frac{y(x)}{3x-y^2(x)}$ ? I'm trying to solve this first order non-linear equation. I've tried to plug in a couple of different things and would appreciate if anyone could point me in ...
2
votes
1answer
27 views

''Differential equation'' with known solution $\sin$ and $\cos$

I am given the following two two equations $f,g : \mathbb{R} \to \mathbb{R}$ are differentiable on $\mathbb{R}$ and they satisfy $\forall x,y \in \mathbb{R}$ $$f(x+y) = ...
0
votes
0answers
16 views

Showing limit of a solution

Assume $w>0$ and $b(t)$ be continuous on $[0,+\infty)$ and integral of $|b(t)|$ is finite on $[0,\infty)$ show that $y''+(w^2+b(t))y=0$ has solution $\phi(t)$ such that ...
0
votes
0answers
21 views

Stability of $a$ implies $\lim _{t\to \infty} x(t)= a$

I have the differential equation $x'=f(x),x\in\mathbb{R}^n$. Let $a$ be a stable point of the differential equation, I want to prove that if $x(t)$ is a solution such that $\forall ...
0
votes
1answer
15 views

EigenFunction for $\frac{\partial f}{\partial t}+f\frac{\partial f}{\partial x} =\frac{2f^2}{x}$

When studying a computer vision problem I end up with a function $f(x,t)$ that satisfying $\frac{\partial f}{\partial t}+f\frac{\partial f}{\partial x} =\frac{2f^2}{x}$. My question includes two ...
-2
votes
0answers
14 views
1
vote
0answers
32 views

Initial Value for an ODE Problem

I have the following ODE $\mathbf{A}\dfrac{d\vec{y}}{dt}+\mathbf{B}\vec{y}=\vec{x}$, where $\vec{y}$ and $\vec{x}$ are $n\times1$ vectors and are functions of $t$, and $\mathbf{A}$ and $\mathbf{B}$ ...
2
votes
1answer
19 views

Fourier Series Coefficient

I am trying to review the basics. Find the Fourier series for the function $$f(x) =\left\{ \begin{array}{l l} 2x & \quad -\frac{\pi}{2}<x<\frac{\pi}{2}\\ 0 & \quad ...
1
vote
0answers
21 views

Solving a homogeneous linear system of differential equations: no complex eigenvectors?

I have to solve the following equation by diagonalization. $ X' = \begin{bmatrix}1 & 1\\1 & -1\end{bmatrix} X$ I was able to determine the complex eigenvalue roots: $det(A-\lambda I)=0$ ...
3
votes
1answer
13 views

How does the recursion relation work in the solution to this differential equation (using series)?

Sorry for the vague title but it would not let me post the first step and last step of this equation (too many characters!). How does $$\dfrac{a_0}{3n(3n-1)(3n-3)(3n-4)\cdots 9 \cdot 8 \cdot 6 \cdot ...
1
vote
0answers
14 views

Nonhomogenous differential equations

I have 2 nonhomogenous differential equations ($\alpha,\beta, c, d$ are constants and $z,y,z_2,y_2$ are functions of x) 1) $z_2'(x)-\alpha[z_2(x)-2z_2(x+c)-y_2(x+c)]=-2\alpha z(x+c)[2y(x+c)+z(x+c)]$ ...
0
votes
1answer
14 views

Does Runge Kutta need future state of system?

In order to use the RK methods, you need to know the state of the system at future time-steps which can be expensive to compute (e.g., in physics simulations). As a simple example I'll use RK-2: In ...
0
votes
3answers
25 views

IVP with Laplace Transform

My attempt: Y = Laplace $$s^2Y -sy(0) - y'(0) - 3Y = ??$$ How do I set up $$h(t)$$ in the form of laplace?
-3
votes
1answer
52 views

Show that $y'=x|y|$ has unique solution

Show that the ODE $$y'=x|y|$$ has an unique solution for all $ (x,y) \in \mathbb R^2$. My attempt: I am not sure if I should consider the two subsets of the domain that sepair $y\leq 0$ and $y ...
0
votes
1answer
40 views

Solutions to a Linear Equation

Let $S$ and $T$ be vector spaces and $L : S\longrightarrow T$ be a linear map. Say $v_1$ and $v_2$ are distinct solutions of the equation $Lx = y_1$ while $w$ is a solution of $Lx = y_2$. In terms of ...
-3
votes
1answer
37 views

Find an integral factor of $3(y+1)dx-2(x-1)dy=0$ [on hold]

What relationship exists between the $h$ and $p$ such that: $$(x-1)^h(y+1)^p$$ is an integral factor of $$3(y+1)dx-2(x-1)dy=0$$
0
votes
0answers
16 views

Modified Airy differential equation

Trying to solve the following differential equation: $y''-kz*y=0$ where k is a constant, I found that the general solution was $y(z)=C_1*A_i(z* \sqrt[3]{\xi} )+ C_2*B_i(z* \sqrt[3]{\xi} )$ on a ...
0
votes
2answers
18 views

ODE ‎Phase portraits of 2x2 systems.‎

Soving a linear 2 by 2 systems yeilds two eigenvector $λ_1= 0$, $λ_2\lt 0$. In this case, do we say the origin is a stable node?
0
votes
1answer
45 views

Solving the ODE $y''-2x^2y'+4xy=x^2+2x+2$ using power series

I am trying to solve this nonhomogeneous ODE: $$y''-2x^2y'+4xy=x^2+2x+2$$ I know it's a power series, but when I get down to the very end, I end up with a $C_0$ term, a $C_1$ term, and a $C_2$ term. ...
2
votes
0answers
37 views

Solution of differential equation $x'=Ax$ where $A=PJP^{-1}$

Let $A$ be a $n\times n$ matrix. Suppose $A=PJP^{-1}$ being $J$ the Jordan form of $A$. Prove that if $x(t)=(x_1(t),\dots,x_n(t))$ is a solution for $x'=Ax$, then ...
0
votes
0answers
45 views

Periodic orbits (Spanish: ORBITAS PERIÓDICAS) [on hold]

Where can I find properties and the classification of limit cycles in the plane? I know there are in Sotomayor's page 228, but there are not so many details [there]. (Spanish: donde puedo ...
3
votes
1answer
21 views

How to find the maximal interval of existence of the solution for the following initial value problem?

Consider the following initial value problem, $$ \dot x = tx^3 \\ x(0) = x_{0} $$ We have the following theorem, . Since the hypotheses of the theorem are satisfied, we must have a solution on ...
1
vote
0answers
28 views

Laplace equation on infinite strip

I'm trying to solve the following problem using the Fourier transform: $$u_{xx}+u_{yy}=0$$ on the domain $\;0\lt y\lt b$ , $-\infty\lt x \lt \infty \;$ with the following conditions: $$ u(x,0)= ...
-1
votes
0answers
18 views

Find a Integral factor [duplicate]

What relationship exists between the h and p thay : enter preformatted text here (x-1)^h(y+1)^p is Integral factor of ...
0
votes
0answers
35 views

Help with first order ODE homework

So I have this question on differential equations, my professor has already explained it to me 3 times but I still do not understand the concept. So I thought I'd try asking here. The question is ...
1
vote
0answers
23 views

General solution for system of differential equations with only one eigenvalue

If I'm given a system of equation of the form $$\begin{cases} \frac{dx}{dt}= ax+by \\ \frac{dx}{dt}= cx+ey\end{cases}$$ I get the general solution finding the eigenvalues and eigenvectors of the ...
1
vote
0answers
18 views

Find the indicial equation

How do i find the indicial equation to $x^2y''+4xy'+4y=12-12x^2$ I used the frobenius method but got stuck: $x^2y''+4xy'+4y-12+12x^2=0$ $x^2(y''+12)+4xy'+4y-12=0$ ...
2
votes
0answers
26 views

Find the 3rd order DE whose general solution is $ y= C_1e^{2x} + C_2\cos x + C_3 x\sin x $

My attempt $$ \begin{matrix} y &=& C_1e^{2x} &+& C_2\cos x &+& C_3x\sin x\\ y' &=& 2C_1e^{2x} &-& C_2\sin x &+ &C_3(\sin x &+& x\cos x)\\ ...
0
votes
0answers
7 views

Reference: Differential operators and principal symbols

I am looking for good references about differential-/pseudodifferential operators and principal symbols. thanks
0
votes
1answer
28 views

Find a 3rd order linear homogeneous differential equation with constant coefficients whose solution is $y=x\sin(x)$

Find a 3rd order linear homogeneous differential equation with constant coefficients whose solution is $y=x\sin(x)$ Here is my attemot so far $$y = x\sin x\\y'=x\cos x+\sin x\\y'' = -x\sin x + ...
0
votes
2answers
42 views

Do solutions of $\dot{x} = \frac{x}{t^2} + t$ exist satisfying $x(0) =0$

Suppose we have the 1-dimensional ODE \begin{equation} \dot{x} = \frac{x}{t^2} + t \end{equation} Do there exist solution curves with initial condition $x(0)=0$? If you proceed in a standard way ...
1
vote
1answer
172 views

What's the name of the differential equation

Consider the ODE $$\frac{y''}{x}+\frac{y}{x}=0,$$ what do we call this equation at the point $x=0$?
1
vote
0answers
22 views

Omega limit set of omega limit set

Suppose $\phi(t;a)$ is a smooth $n$-dimensional flow in $\mathbb{R}^n$ and $A \subset \mathbb{R}^n$. The omega-limit set of $A$ is defined to be $\omega(A):= \bigcup_{a \in A} \omega(a)$, where ...
1
vote
1answer
20 views

second order homogenous linear ODE with variable coefficients [on hold]

Let $y_1(x)$ and $y_2(x)$ form a complete set of solutions to the differential equation $$ y"-2xy'+\sin(e^{2x^2})y=0,$$ $x \in[0,1]$ with $y_1(0)=0, y_1'(0)=1, y_2(0)=1, y_2'(0)=1$. Then Wronskian ...
1
vote
1answer
26 views

differential inequality involving the square of the function

It is written in a book, (Bertozzi- Majda, vorticity and incompressible flow page 106) that given a differential inequality of the following type: $ \frac{d}{dt}\|u^{\epsilon}(t)\| \leq ...
0
votes
1answer
18 views

Logistic differential equation to model population

Problem Description: The population of the world was about 5.3 billion in 1990. Birth rate in the 1990s ranged from 35 to 40 million per year and death rates ranged from 15 to 20 million per year. ...
1
vote
0answers
9 views

Two questions regrading to Laplace equation, the Green's Reconstruction Formula

All the following we use Evans notation. By Green's reconstruction formula, we could represent $u$ by $$ u(x)=\int_\Omega-\triangle u(y)G(x,y)dy-\int_{\partial \Omega}u(y)\partial_\nu ...
0
votes
0answers
21 views

Using the definition of the matrix exponential to solve a 2x2 system of DE's

I've got a system of equations: x1' = x2 x2' = -k2x1 And am looking to use the definition of exp(At), where A is the coefficient matrix of the above system, to show that exp(At) = Icos(kt) + ...
1
vote
1answer
17 views

Laplace equation in polar coordinates

Solve the Laplace equation in polar coordinates $u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta}=0$ within the domain $0<\theta<\pi, 1<r<2$ subject to boundary conditions ...
1
vote
0answers
29 views

Measurability of solution of diffusion equation in sub-sigma algebra

I want to solve the following problem: Get $\omega \in \Omega \subset \mathbb{R}$, $x \in D \subset \mathbb{R}^2$ and $0<a_i\leq a(.,.)\leq a_x<\infty$. Let $a( x;. )$ and $f(x;.)$ be ...
0
votes
1answer
21 views

reduction of order; not getting right answer

Use reduction of order or formula 5 to find a second solution $y_2(x)$ Formula 5 is $y_2=y_1(x)\int \frac{e^{-\int P(x) dx}}{y^2_1(x)}dx$ where $P(x)$ is $y''+P(x)y'+Q(x)y=0$ The ODE is $y''+16y=0$ ...