Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.
1
vote
0answers
14 views
What is the difference between an implicit ordinary differential equation and a differential algebraic equation?
I'm rather confused on this particular point. What is the difference between an implicit ordinary differential equation of the form:
x' = f(x',x,t);
and a ...
2
votes
0answers
65 views
Why this synchronization error dynamic for Krasovskii-Lyapunov?
I am attempting to work through "Shahverdiev, Sivaprakasam, and Shore (2002) Lag synchronization in time-delayed systems", but I'm missing something basic up front.
The problem is to take a ...
3
votes
1answer
44 views
Polynomial differential equation
I came across this problem in an old olympiad paper (Putnam?)
Find all polynomials $p(x)$ with real coefficients satisfying the differential equation
$7\dfrac{d }{dx } [xp(x)]=3p(x)+4p(x+1)$ $\ \ ...
4
votes
3answers
97 views
Where to start when learning math (again)?
I have a few questions I hope you can help me answer.
First, I'll introduce myself. I'm a finance undergraduate student in Australia, but I'm originally from Norway. Throughout school I always loved ...
2
votes
4answers
65 views
initial value problem: y'' + 4y = f(t) , y(0)= y'(0)=0. f(t) = { 0 if t <3; t if t >3}
Solve the initial value problem:
$$y'' + 4y = f(t) , y(0)= y'(0)=0. $$
where
$$ f(t) = \begin{cases} 0 &t < 3 \\ t & t > 3\end{cases} $$
I've solved for the homogeneous equation, $y'' ...
2
votes
3answers
82 views
General solution of differential equation of order 3
Please ,how to find that the general solution of $u'''(t)=e(t) , t\in [0,1]$ is given by
$u(t)=c_0+c_1t+c_2 t^2 +\frac12 \int_0^t (t-s)^2 e(s) ds$
$e:(0,1)\rightarrow \mathbb{R}$, and $e\in ...
3
votes
1answer
68 views
How to solve the two dimensional Laplace's equation for certain cases?
Had a doubt regarding Laplace's equation.
In many textbooks, the general solution to the two dimensional Laplace equation is mentioned as:
$$\Phi(\rho,\phi) = A_{0} + B_{0}\ln(\rho) + ...
0
votes
3answers
69 views
Does this IVP have a unique solution for all $x \in \mathbb R$
Is $\displaystyle {dy\over dx}=\sin(y)$ with initial conditions $y(X)=Y$ guaranteed to have a unique solution for all $x\in\mathbb R$?
0
votes
1answer
29 views
Grand Prix Race- Differential Equations
Driver A has boon leading archrival B for a while by a steady 3 miles. Only 2 miles from the finish, driver A ran out of gas and decelerated thereafter at ta rate proportional to the square of his ...
2
votes
0answers
25 views
Looking for online matlab-based differential equations course/text.
I am looking for an online ODE course that would be matlab/project-oriented. A full online text/course in the spirit of this linear algebra text is preferred.
I know about the following
CODEE and ...
2
votes
4answers
50 views
Differential Equations Reference Request
Currently I'm taking the Differential Equations course at college, however the problem is the book used. I'll try to make my point clear, but sorry if this question is silly or anything like that: the ...
3
votes
0answers
50 views
Non-linear first order differential equation
I've found this particular equation rather tough, can you give me some hints on how to solve
$$\dot{y}+t\cos(\frac{\pi}{2}y)+(1-t)=y$$
Thanks a lot.
0
votes
0answers
19 views
The meaning of a boundary value problems at resonance?
What is the meaning of a boundary value problems at resonance? For example,
\begin{eqnarray}
x''(t)=f(t,x(t))\\
x'(0)=0,\ x(1)=x(\eta),\ \eta\in (0,1)
\end{eqnarray}
The boundary value problem is ...
2
votes
0answers
43 views
What is the physical meaning of fractional calculus?
What is the physical meaning of the fractional integral and fractional derivative?
And many researchers deal with the fractional boundary value problems, and what is the physical background?
What ...
1
vote
2answers
34 views
Solving System of Differential equations
The general solution to differential equation
$$x'=Ax$$ where A is a square matrix is given by solving for the eigenvalues and then eigen vectors of matrix $A$. However, is there a general method if I ...
2
votes
2answers
33 views
Finding a strong enough solution to a specific PDE problem.
Let $U\subset \mathbb{R}^n$ with smooth boundary $\partial U$. And consider the expression
$$\Delta u = f.$$
$$\text{+"convenient boundary conditions"}$$
In my specific case $f\in H^2_0$. Under ...
2
votes
1answer
149 views
Description of a circle in vectors
Letting $\vec{r}=\begin{pmatrix}x(\theta)\\y(\theta)\end{pmatrix}$ and that $\vec{v}^\perp$ denote a vector perpendicular to the vector $\vec{v}$. Then are there values of $c,d$, with ...
1
vote
0answers
39 views
optimal control -Taylor expansion - PDE problem
I am trying to follow perturbation analysis in this paper (Optimal control of fluid limits of queuing networks and stochasticity corrections) and I am stuck at one point.
For the given control ...
1
vote
1answer
31 views
Simple ODE question…
I've run up on something that is intuitively obvious but I am having a little trouble seeing formally. Could someone show me exactly why $y^\prime=-2+t-y$ is asymptotic to $y=x-3$. Thanks, in advance ...
1
vote
1answer
47 views
worked conservative question. Can someone double check?
Can you please check this question for me because my answer is different to my friends and they say it's wrong...
Given the equation: $G = (x^3 -3xy^2)\vec{i} +(y^3-3x^2y)\vec{j} +2\vec{k}$ is ...
0
votes
0answers
14 views
Existence and uniqueness of a matrix differential equation
I came across the following differential equation:
$$
N'_x(x,z)=G(x,z)N(x,z)
$$
where $N(x,z)$ is a $2\times2$ complex matrix with variables $x$ and $z$, $N'_x$ is its derivative with respect to ...
1
vote
0answers
17 views
Lipschitz constant for $F(t,y,z)=(z,f(y,z)\sin t)$
Let $f\in C^1(\mathbb R^2,\mathbb R)$.
Prove that all solutions for $x''=f(x,x')\sin t$ such that $x(0)=x(2\pi)$ and $x'(0)=x'(2\pi)$ have period $2\pi$.
I'm in the process of solving the above ...
0
votes
0answers
24 views
The variables have been separated?
For the differential equation $p(x)+q(y(x))y'(x)=0,$ I write it in the form $\int p(x)\,dx+\int q(y)\,dy=C$ where $C$ is an arbitrary constant and $y=y(x)$. Is it I can say that "in $\int ...
0
votes
1answer
37 views
Ordinary differential equations with double resonance
I want to know what is the definition of "resonance, double resonance" in
ordinary differential equations with double resonance
Please,
Thank you.
2
votes
0answers
29 views
Closed form of the solution of a nonlinear differential equation
I should solve the following problem: given a function $u(x)$, the sum of the function and its reciprocal must be equal to the integral of the function raised to $k$. Taking the derivative of the two ...
2
votes
0answers
46 views
Method of undetermined coefficients for the input functions associated with the unit step
I am trying to solve a second order non-homogeneous differential equation where $x(t)$ has $u(t)$, the unit step as a part. i.e. $ x(t)= f(t)u(t) $
I know how to 'guess' the particular solution for $ ...
-1
votes
0answers
21 views
Proving an operator is self-adjoint
Prove the operator $L$ is self-adjoint.
$$L: y''-q(x)y(x)=-\lambda y(x) , x\in[0,\pi],$$
$$\lambda(y'0)-hy(0))=h_{1}y'(0)-h_{2}y(0),$$
$$\lambda(y'(\pi)+H y(\pi))=H_{1}y'(\pi)+H_{2}y(\pi),$$
where ...
0
votes
1answer
176 views
Linear birth death process, probability of extinction by time t
I have a linear birth death process with birth rates $\lambda n$ and death rates $\mu n$ .
Let r(t) be the probability of extinction by time t.
If there is 1 individual alive at time 0 explain why
...
0
votes
1answer
38 views
Question about eigenvalues
I have this :
i dont understand why they write $\lambda=m^2 , m\in \mathbb{N}\cup\lbrace0\rbrace$ ,
it's right that $\lambda=m^2$ is the eigenvalues of $(P_0)$ ,but $0$ is not an eigenvalue !.
...
1
vote
2answers
24 views
spherically symmetric configurations
$$\Delta S -S +S^3=0$$ How this Differential equation can be written in this form:
\begin{equation}
\frac{d^2S}{d\rho^2}+\frac{D-1}{\rho}\,\frac{dS}{d\rho}
-S+S^3=0
\end{equation}
Which is ...
1
vote
3answers
142 views
Sixth order differential equation
Find the general solution of $y^{(6)}+2y^{(4)}+y'' = 0$.
$r^6+2r^4+r^2=0$
$r^2(r^4+2r^2+1)=0$
$r^2[(r^2+1)(r^2+1)]$
So we have the roots:
$0$: Multiplicity 2
$+i$: Multiplicity 2
$-i$: ...
1
vote
1answer
35 views
PDE initial value problem
Show that the solution of the initial value problem for
$u_t+u_x=\cos ^2 u$
is given by
$u(x,t)=\tan^{-1} \{ \tan [u_o(x-t)]+t\}$,
where $u_0(x)$ is the initial condition.
My attempts at a ...
0
votes
0answers
30 views
Find a differential equation the difference scheme is consistent with
Given the following scheme:
$\frac{1}{2}(u^n_{j+1}+u^n_{j-1})-\frac{1}{2}\frac{\Delta t}{(\Delta x)^3}(u^n_{j+2}-2u^n_{j+1}+2u^n_{j-1}-u^n_{j-2})$
how can I find a differential equation that's ...
0
votes
1answer
21 views
What functions are solution to a homogeneous system of differential equations?
Given a vector $\vec{u} \in \mathbb{R}^n$. For what functions $\psi(t)$ can $\vec{x}(t) = \psi(t)\vec{u}$ be a solution of $\dot{\vec{x}} = A \vec{x}$ for some $n \times n$ matrix $A$?
I'm trying to ...
1
vote
0answers
17 views
How to solve two-level Schrödinger equation using Floquet theorem?
Consider a sinusoidal driving two-level system:
$$
i \left(
\begin{array}{c}
\dot C_1(t) \\
\dot C_2(t) \\
\end{array}
\right)=\left(
\begin{array}{cc}
-\frac{\omega _0}{2} & ...
0
votes
1answer
37 views
Linear Differentiation
I have to determine whether there is normal linear differentiation equation $a_2(x)y'' + a_1(x)y' + a_0(x)y = 0$ on $\mathbb{R}$ such that $u_1, u_2 \in C^2(\mathbb{R})$ defined by $u_1(x) = x, u_2(x) ...
3
votes
2answers
58 views
If $f$ is even and $y'=f(y)$ then $y$ is odd
Let $f\in C^1(\mathbb{R}, \mathbb{R})$ be an even function.
Consider the maximal solution $y\colon\left]\alpha ,\beta\right[\to \mathbb{R}$ of the IVP $$y'=f(y),\ y(0)=0$$
Prove that $y$ ...
0
votes
1answer
45 views
Green's function. Basic
Can anyone give some advice about books where I could find introductory information about Green's function. What are the methods of constructing Green's function.
Actually, Green's function for 3D ...
9
votes
3answers
160 views
Examples of nonlinear ordinary differential equations with elementary solutions.
I am looking for nice examples of nonlinear ordinary differential equations that have simple solutions in terms of elementary functions. (But are not trivial to find, like, for example, with ...
1
vote
1answer
32 views
How can you prove Euler's phase angle formula for differential equations?
How can you prove this formula:
$C_1 e^{(\alpha + i\beta) t} + C_2 e^{(\alpha - i\beta)t}=Ke^{\alpha t}\cos {(\beta t + \phi)}$
This gives $x(t)$ in the second-order differential equation for an ...
2
votes
1answer
258 views
Runge-Kutta 4 - solving system of 6 differential equations (BVP)
I'm facing a tricky problem. I need to solve a system of 6 differential equations numerically, but I don't have 6 IVP (initial value problem) conditions, instead I have 6 BVP (boundary valye problem) ...
1
vote
1answer
23 views
System of differential equations: Inverse matrix of a fundamental matrix
I'm trying to show:
Let $A:[0,\infty[\to \mathcal{M}(n,\mathbb{R})$ a function and suppose that all solutions of the system of differential equations:
$$\vec{x'}(t)=A(t)\vec{x}(t) \ \ \ (\star)$$
...
5
votes
1answer
137 views
Tough Inverse Fourier Transform
In reference to this answer I gave the other day, I came across a very interesting function whose IFT would be nice to evaluate as part of completing the solution to the problem I answered. The ...
1
vote
0answers
20 views
Derivative methods for artifical neural networks with single hidden layer
I am trying to optimize the output of a given neural network with a single hidden layer. To accomplish this, I intend to find solve for all combinations of inputs where the derivative of the neural ...
1
vote
1answer
41 views
Second order nonlinear delay differential equation
I have to solve the following delay differential equation:
$$\ddot{x}(t)+A\sin(\omega x(t-\tau))=0$$
Can someone give me a hint on how to solve this equation?
Thanks
1
vote
2answers
52 views
Hard time figuring out word problem
Computers are beginning to lose power. The number of computers losing power doubles every minute. At 10 minutes half of the computers have lost power. After how many minutes will all computers be ...
2
votes
1answer
44 views
Questions on differential equations of matrices
I have a differential equation $$N'_x(x)=G(x)N(x)$$ where $N, G$ are $2\times2$ matrices depending on $x$, and $G$ satisfies $\sigma G+G\sigma=0$, $\sigma$ is one half of the pauli matrix, i.e. ...
6
votes
2answers
204 views
Inverse Laplace Transform help
Is the information below correct?
Find the inverse Laplace transform of $$ F(s) = \frac{s}{s^2 + 4s + 13}$$
Soln:
a) Complete the squares to simplify our denominator
$$ s^2 + 4s + 13 = (s+2)^2 + 9 ...
1
vote
0answers
46 views
Lyapunov Stability of Non-autonomous Nonlinear Dynamical Systems
Let $\mathbf{F}:X\times\mathbb{R}^{+}\to X$ be a non-autonomous dynamical system, which is governed by $\dot{\mathbf{x}} = \mathbf{F}(\mathbf{x}, t, u)$, viz,
\begin{equation}
\begin{split}
\dot{x}_1 ...
2
votes
1answer
265 views
Going in the direction of the gradient
First, a motivating example. Suppose $f(x)$ is convex, differentiable, with a single minimum $x^*$. Then the differential equation $$\dot{x}(t) = -\nabla f(x(t))$$ drives $x(t)$ to $x^*$.
Now my ...
