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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2answers
152 views

Symbol Of Differential Operators

When we are given a differential operator of the form $Lf : = \sum_\alpha a_{\alpha}(x) D^ \alpha f(x) $ , we can define the symbol associated with it to be the function: $a(x,y) := \sum_\alpha ...
0
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1answer
256 views

Steady-state solution of an ODE

This is the problem given: I am not entirely sure what my Professor expected from an answer, but it seems I am to find the coefficient, angular frequency, and phase of the non-homogenous solution ...
0
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1answer
104 views

Finding the spectrum of the Schrodinger operator

Let $H(f) = -f'' + V(x) f$ be the Schrodinger operator on $\mathbb R$. I am trying to calculate the spectrum (eigenvalues) of the operator $H$ in $L^2(\mathbb R)$ for various choices of $V$. In ...
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0answers
153 views

Lipschitz Questions

I want to ask one general question, and after that I would like to know if my method is correct (for determining whether a function is Lipschitz with respect to y) Is the following statement true? ...
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0answers
57 views

Second order equations on manifolds

In my notes, the lecturer considers a smooth vector field $v: TM\to T(TM)$, with $M$ a smooth manifold. Let's write $$v(u,e)=((u,e), (a(u,e),b(u,e)).$$ It is said that $v$ is a second order equation ...
1
vote
1answer
52 views

Let $y=(y_1,y_2)^t$ be a solution of the system of$ y'_1=y_2$,$y'_2=ay_1+by_2$.Then every solution $y(x)\rightarrow 0$ as $x\rightarrow \infty$ if

I was thinking about the following problem that says: Let $a,b\in \mathbb{R}$.Let $y=(y_1,y_2)^{t}$ be a solution of the system of $y'_1=y_2$,$y'_2=ay_1+by_2$. Then which of the following options is ...
0
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1answer
855 views

Stability of critical points for systems of ODEs

Suppose we have: $\frac{dx}{dt} = 14x-\frac{1}{2}x^2-xy$ $\frac{dy}{dt} = 16y-\frac{1}{2}y^2-xy$ My textbook outlines the following steps: For the critical point (0,32) of the above system, ...
0
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1answer
51 views

Find the linearization of the following system

$f(x,y) = 3x-x^2-xy$ $g(x,y) = y+y^2-3xy$ Find the linearization of the system at the critical point (1,2). I know the Jacobin can be used for this problem, but I was wondering if it's permissible ...
2
votes
2answers
236 views

Solving the time-independent Schrodinger equation for particle in a potential well

I'm solving a quantum mechanics problem for the particle in a potential well, and the equation I have to solve is $$\frac{d^2\psi}{dx^2}+k\psi=0$$where $$k=\frac{2mE}{\hbar^2}$$ This seems easy enough ...
5
votes
0answers
70 views

Solution to differential equation $f^{(n)}-(n+1)f^{(n-1)}-(n+1)nf^{(n-2)}-\dotsc-(n+1)!f=g$

Let $n$ be a given positive integer and $g$ be a continuous function. We are looking for a function $f \in C^n(\mathbb{R})$ such that $$f^{(n)}-(n+1)f^{(n-1)}-(n+1)nf^{(n-2)}-\dotsc-(n+1)!f=g.$$ It ...
2
votes
1answer
205 views

Periodic solution of nonlinear differential equation

Let $C$ be a positive constant. Consider the following system of differential equation with inial value \begin{eqnarray} z(t)+\frac{\sqrt{2}}{2} u'(t)-1=0 \\ 2u''(t)+C \sin(2u(t))=0 \end{eqnarray} ...
0
votes
1answer
194 views

Confused about Lipschitz functions

Given a function $f(x,y)$, I want to show whether the function satisfies a uniform Lipschitz condition with respect to $y$, and determine the Lipschitz constant $L$. Questions: 1.) I know that ...
0
votes
2answers
210 views

Show that the following cycle has a limit cycle

By direct calculation show that (using polar coordianted) that $$ \dot x=x-y-x(x²+y²) $$ $$ \dot y=x+y-y(x²+y²) $$ Show that this has a limit cycle I need help understanding how to test whether it ...
0
votes
3answers
221 views

General Solution for a given system of equations

Find the general solution of this system of equations: $$x' = \pmatrix{-1&0&0\\1&0&-1\\1&1&0}x$$ I got the eigenvalues to be: $\lambda = -1,\pm i$ The eigenvectors ...
0
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1answer
525 views

Which of the following is gradient/Hamiltonian( Conservative) system

The question that I have to solve is found below. However, I do not know how to start the solution since I am unsure about the defintion of a Gradient/Hamiltonian System. What must I check first to ...
0
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2answers
282 views

Fixed points of a nonlinear system

Find the fixed points of a nonlinear two-dimensional system: $$\dot{x} = \sin y$$ $$\dot{y} = x - x^3.$$ I know that $0 = x(1 - x²) \implies x = 0, 1, -1$. I am not sure what to do after this.
4
votes
3answers
435 views

Differential equation, Stability , Lyapunov function

Given a system of differential equations \begin{eqnarray} x'&=&2y(z-1)\\ y'&=&-x(z-1) \quad (1)\\ z'&=&xy \end{eqnarray} Note that $u_0$=(0,0,0) is an equilibrium point of the ...
1
vote
1answer
49 views

General solution differential equation

Find the general solution: $$x'= \pmatrix{-2&-1\\2&0}x$$ I got the eigenvalues to be $\lambda = -1 \pm i$. Now finding the eigenvectors: $(\lambda = -1 + i )= v_1 =\pmatrix{-2 -(-1 ...
6
votes
3answers
193 views

How do I show there are no elementary function solutions for the differential equation $f''(x)=f(\sqrt{x}), x>0$?

How do I show there are no elementary function solutions for the differential equation $f''(x)=f(\sqrt{x}), x>0$ in the $C^2(0,\infty)$ space solutions?
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2answers
66 views

Solution of the problem $y’(t)=f(t)y(t), \; y(0)=1$ where $f:\mathbb{R}\to\mathbb{R}$ is continuous

Consider the initial value problem $$y’(t)=f(t)y(t), \;y(0)=1$$ where $f:\mathbb{R}\to\mathbb{R}$ is continuous. Then this initial value problem has: Infinitely many solutions for some $f$. A ...
3
votes
2answers
2k views

How can the Laplace transform be used to solve piecewise functions?

For example, suppose we have the following two problems that we'd like to find the Laplace transform of: $f(t) = \begin{cases} 1, & t \lt 2 \\ 0, & t \geq 2 \end{cases}$ $f(t) = ...
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vote
1answer
1k views

Find the bifurcation points for the following system of ODEs

I'm trying to find the equilibrium points for the following system: \begin{align} \frac{dx}{dt} &= x-xy \\ \frac{dy}{dt} &= x+a-y^2 \end{align} For $a = -1.5,-1,-0.5,0,0.5,$ and $1$. I know ...
0
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1answer
289 views

What does it mean for a phase portrait to have “limit cycle behavior?”

Consider a system: $dx/dt = x(1-x)-\frac{kxy}{kx+1}$ $dy/dt = ry(1-\frac{y}{x})$ For values of r as 0.15, 0.11, and 0.05, which of the corresponding phase portraits displays limit cycle behavior? ...
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1answer
176 views

How can I use the translation theorem to show that two inverse Laplace transforms are the same?

Use partial fractions to find the inverse Laplace transform of $F(s) =\large \frac{s+1}{s^3+5s^2+6s}$. Then use $u = s+1$ to show that $F(s) = G(u) = \large\frac{u}{(u-1)(u+1)(u+2)}$. Use partial ...
1
vote
1answer
97 views

Bivariate Legendre polynomials?

We have $\frac{1}{\sqrt{1+x^2}} = \sum^{\infty}_{n=0} P_n(0)x^n$ where $P_n(x)$ is a Legendre polynomial of degree $n$. Is there something similar for two dimensions i.e. $\frac{1}{\sqrt{1+x^2+y^2}}$ ...
2
votes
1answer
95 views

What is the correct way to verify solutions from different ode symbolic solvers against each others?

I am not a math major, so I thought to check with the experts here on this. Given some ODE $y'(x)=f(y,x)$ solved by CAS system A which gives answer $y_1(x)$. Same ODE solved by CAS system B with ...
0
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1answer
414 views

Laplace Transform Dirac Delta Function

Find the solution of the initial value problem. $y'' +2y' +2y = \delta(t - \pi)$ with initial conditions $y(0) =1, y'(0) =0$. What I did was take the Laplace and got: $(s^2Y(s) - s) + ...
3
votes
1answer
88 views

Solve $\partial^{2}_{x} \left[x^{2}p\right] + \partial_{x} \left[\left(x-1\right)p\right]$

How do I solve the following differential equation? $\partial^{2}_{x} \left[x^{2}p\right] + \partial_{x} \left[\left(x-1\right)p\right] = 0$ I tried a Fourier transform which leads to ...
3
votes
1answer
278 views

Generic Sturm-Liouville Problem

The equation $\ddot{y}+(\lambda(r)^2-L^2r^{-2})y=0$ seems that can be cast in th eform of a generic Sturm-Liouville problem as $-\ddot{y}+q(r)y=\lambda_0\lambda(r)^2$ with $q(r)=L^2r^{-2}$ It can ...
0
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1answer
122 views

Application of Backward euler method

Information Let $y'(t)=f(t,y(t))$ and $y(0)=y_0$ The backward euler method together with the center rule is given by: $y(t_k)=hk$ where $h\in (0,\frac{1}{K})$ is the step size. Recursion: ...
2
votes
1answer
274 views

Can I differentiate both side of a differential equation?

This is a trivial question, please note I'm not a professional in this environment, I'm just learning. Let's suppose I've got this simple eqution: $L\frac{d i(t)}{dt}=-\frac{1}{C}\int i(t) dt$ I ...
2
votes
2answers
303 views

Solving differential equation with power series

$$\begin{cases} w''=(z^2-1)w \\ w(0)=1 \\ w'(0)=0 \end{cases}$$ I tried the following: Let $$w(z)=\sum_{j=0}^{\infty}w_j z^j$$ $$\implies w''(z)=\sum_{j=0}^{\infty}j(j-1)w_j z^{j-2}$$ $$\implies ...
0
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1answer
196 views

Omega limit set of a system

The system of ODE is $$\begin{cases} \dot{x}=-y(1-x^{2}) \\ \dot{y}=x+y(1-x^{2}) \end{cases} \tag{$\ast$}$$ Claim: $\forall p\in\left\{ (x,y)\in\mathbb{R}^{2} : |x|<1,\ x^{2}+y^{2}>0\right\} ...
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1answer
136 views

Asymptotic stability of fixed point

$f'(t)=af(t)(K-f(t))-bf(t)g(t)$ for $a,b,c,d,t,K>0$ $$g'(t)=cf(t)g(t)-dg(t)$$ This system has 3 fixed points (You can evaluate them if you set the 2 equations = 0). One point is ...
0
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1answer
383 views

Integrating factors — how in the world does one calculate those?

Is there an easier way of computing an integrating factor for differential equations? I need help understanding how to calculate those. I know the reason for them but just not familiar with how to ...
2
votes
2answers
82 views

Need to explain an application of $y' = ky$

I have to write application of the ODE $y' = ky$. i want application or any that explain little bit. Thanks!
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2answers
81 views

What do I need to know to simulate many particles, waves, or fluids?

I've never had a numerical analysis course so I don't know what I need to know. I'm just wondering what kind of books I should get to make me able to simulate these things. I'm wanting to simulate ...
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1answer
138 views

How can I solve for the general solution of a second ODE using Laplace transform?

I am interested in solving for the most general solution, in other words with the constants c1, c2, etc. Take the first example from Paul's Notes: ...
2
votes
1answer
98 views

Differential Equations background

What are the prereqs for differential equations? Do you need to know integral calculus too, and if so, to what extent? I want to learn about DE's as quick as possible but I'm not sure if I'm ready ...
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1answer
56 views

Differential Equation and Stability

I have an equation: $V_{t+1}=V_t+r(S(V_t))$. r is a constant when$(r=?)$ is $V$ asymptotically stable and when otherwise? What I tried is, finding equilibrium points, I got: $S(V_t)=0 $ and $r=0$. ...
0
votes
2answers
57 views

Differential equation : $u' = \sin(t)\exp(u)$

I have a question about taking the absolute value of an argument when working with logarithm. I found this solution $u(t) = -\log(\cos(t)+C)$. But i am not sure if I have to take ...
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1answer
205 views

Repeated Root Eigenvalues

The question is: Solve the initial value problem: $$\begin{align*} \frac{dx_1}{dt}&=40x_1-6x_2+18x_3,\\ \frac{dx_2}{dt}&=-6x_1+45x_2+12x_3,\\ \frac{dx_3}{dt}&=18x_1+12x_2+13x_3,\\ ...
2
votes
1answer
224 views

Kummer's Equation

I'm trying to show that Kummer's equation can be solved by deriving $$xL_{n}^{''}(x)+(1-x)L_{n}^{'}(x)+nL_{n}(x)=0$$ from the Laguerre polynomials: ...
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0answers
65 views

Solution to matrix ODE $Ay'[x] + B\frac{y[x]}{x} + Cy[x] = 0$?

Does there exist a closed form solution to the homogeneous system of ODEs $$Ay'[x] + B\frac{y[x]}{x} + Cy[x] = 0,$$ where $A$, $B$, and $C$ are $n$ x $n$ (constant) matrices, and $y$ is an ...
0
votes
1answer
88 views

Solving a differential equation

I need to solve the integral $$\int^x \frac{1}{t^2}e^{t^2}dt. $$ The answer should be $$\sqrt{\pi}\left(\text{erfi}(x)\right)-\frac{e^{x^2}}{x} + C$$ What does $\text{erfi}(x)$ mean? Can anybody ...
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vote
1answer
50 views

How is $x = 0$ a solution to x' = Ax?

This may seem obvious, but how would you explain something like this? This is referring to the homogeneous system of differential equations, originally from $$x'(t) = A(t)x(t) + b(t)$$ where $b(t) = ...
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vote
2answers
77 views

Eigenvalues and IVPs

So I have this question: Solve the initial value problem: $$\begin{align*} \frac{dx_1}{dt}&=3x_3-2x_4,\\ \frac{dx_2}{dt}&=-2x_3+3x_4,\\ \frac{dx_3}{dt}&=3x_1-2x_2,\\ ...
2
votes
1answer
160 views
2
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4answers
227 views

Is there any research in mathematical biology that isn't heavy in differential equations?

I'm near the end of my pure math undergrad trying to decide what sort of math I'm interested in for graduate school. I've always thought the idea of mathematical biology was cool, but it seems like a ...
0
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1answer
102 views

Prove that the ODE above has a special solution of the form $x(t) = ce^{\alpha t}$

Consider an $n^{th}$ order linear differential equation of the form $a_nx^{[n]} +a_{n−1}x^{[n−1]} +···+a_1x′ +a_0x=f(t)$, $(1)$ where the $a_k$ are constants and $x^{[k]}$ denotes the $k^th$ ...