Tagged Questions
0
votes
1answer
52 views
Liouville's formula
I have some questions concerning a proof of Liouville's formula:
$$W'(t)=\text{tr}(A) W(t)$$ where $W$ is the Wronskian of the homogenous ODE.
If the vectors in the columns of the fundamental matrix ...
2
votes
2answers
29 views
Are there real numbers a and b such that $f(x,y,t) = x^a t^b$ satisfies the heat equation?
The question is in the title. The heat equation is as follows:
$$
\frac{\partial f}{\partial t} = k \left( \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} \right),\quad ...
1
vote
0answers
113 views
“Two-speed” linear integro-differential equation
Working on a problem of many-electron dynamics in quantum dots I have arrived to an a following integro-differential equation:
$$\frac{\partial}{\partial t} F(x,t)= - i (x+ v_1 t) F(x,t)-\alpha^2 ...
0
votes
1answer
23 views
proof a property of a linear differential equation
I am supposed to proof that if a map $A:I \rightarrow \mathbb{C}^{\text{nxn}}$ where $I$ is a compact interval is continuous and A satisfies $\forall x \in \mathbb{C}^{\text{nxn}}: Re\langle Ax,x ...
1
vote
1answer
29 views
Uniqueness of first order differential equation?
We have a theorem that says: Let $h: I\rightarrow \mathbb{R}$ and $g:J \rightarrow \mathbb{R}$ be continuous functions, $t_0 \in I $ and $y_0 \in \text{int(J)}$, then the differential equation ...
0
votes
1answer
27 views
Let $F(x,y,z)=\frac{-c r}{||r||^3}$ and $r = \langle x,y,z \rangle$
Compute $ \frac{ \partial F1}{\partial y}$ & $\frac{ \partial F2}{\partial x}$.
How do I do this if $F(x,y,z) = \frac{-cr}{||r||^3}$ is one function and not a vector of $<.F1.,.F2.>$?
1
vote
1answer
30 views
Derivative of solution of ODE
I have a set of nonlinear differential equations with parameters.
$$\dot{\vec{x}} = F(\vec{x},\vec{\beta}) $$
where $\vec{x} \in \mathbb{R}^p$ and $\vec{\beta} \in \mathbb{R}^q$ ($p,q \in ...
1
vote
1answer
43 views
Simple Phase Diagram Question for a Simple system
Say I have the following system:
$$
y'_1 = -2y_1 + 2
$$
$$
y'_2 = -3y_2 + 6
$$
The solutions being
$$
y_1 = C_1e^{-2t}+1
$$
$$
y_2 = C_2e^{-3t}+2
$$
So for the phase diagram I plot a $y'_1$ and a ...
1
vote
0answers
40 views
Cohomologies of $\mathbb R^n$ with rational differential forms
We can consider de Rham complex $0 \to \Omega^1 \to \Omega ^ 2 \to...$ on $\mathbb R^n$, where $\Omega ^r$ are $r$-forms on $\mathbb R^n$ with rational coefficients. What are homologies of this ...
1
vote
1answer
34 views
Solving differential equation for x
I have a field $\phi(x,t)=\sin(t+|x|)(\frac{x}{|x|})$ where x is a point vector and t is the current time.
If this field describes the acceleration of a particle at a point in space and time:
...
4
votes
0answers
62 views
Optimizing a functional with a differential equation as a constraint
I am working on solving the following optimization problem. I think it is well-poised but, if not, please give me some pointers that could make the question make more sense.
We have a parametric ...
0
votes
0answers
23 views
Derivative of this function and trajectories of a related ODE?
These are some questions from a reply by a deleted user. Since he will probably not be able to comment there, please excuse me for creating this post.
Let $\Phi: \mathbb R^d \to \mathbb R$ be a ...
1
vote
1answer
73 views
Why can't gradient systems have closed orbits?
I've seen the proof provided in Strogatz where he compares the change in the gradient function $V$ after one period $T$; on one hand, $\Delta V=0$ because $V(x(T)=V(x(0))$, while on the other, $\int ...
0
votes
3answers
59 views
Solving a differential equation with more than one dependent variable
It's been awhile since I took differential equations. Now I am using differential equations in another class. This is why you shouldn't sell back books from your major courses. :)
How would I solve ...
0
votes
0answers
28 views
is there any iterative approach to find x and y
$f_{1}(x,y)$ and $f_{2}(x,y)$ are differentiable functions on x and y. Need to find $(x^*,y^*)$ which satisfy below equations.
\begin{eqnarray}
...
0
votes
1answer
53 views
Poincaré map corresponding to submanifold
I am interested in how to calculate the Poincare map corresponding to a submanifold.
The vector field $f(x,y)=(-y,x)$ has the periodic solution $(cos(t),sin(t))$
Now I'd like to compute the Poincare ...
0
votes
1answer
28 views
Information needed about Local Extremas in differential calculus
We know a function $f \in C^2(R)$ has a Local Maximum in the origin $(0,0)$. What can you say about the differential: $d_{(0,0)}^2f(1,-1)<0$?
I've recently got this on a test and I'm not sure if ...
4
votes
0answers
155 views
How to solve this differential equation for $y$ in terms of $x$ and $k$
$$yy'+\frac yx+k=0$$
How to solve this differential equation for $y$ in terms of $x$ and $k$ where $k$ is a parameter of $x$?
$y(x)=y$ is a function and $x(k)=x$ is a gamma function
3
votes
1answer
52 views
How to determine phase image
I need to sketch the phase image belonging to the following vector field (I'm sorry, I don't know the exact terms in English, so I have just freely translated them - thanks for sharing the correct ...
2
votes
2answers
87 views
Particular Solution of Differential Equations
While solving second order non-homogenous differential equations of the form
$y''+y'=5$, I realized that unlike while solving the ones of the form $y''+y'+y=Ax^n$ where we assume, ...
1
vote
0answers
111 views
How simple is it to solve this Differential Equation
How to solve this Differential Equation?
How simple is it to solve this Differential Equation?
Any guidelines? Any hint? How to approach the solution?
Have anybody seen things like it before? ...
$$
...
0
votes
1answer
86 views
Clear intuition for continuity and limit in multivariable functions
I have problem understanding limit and contuinity of a multivariable function. Could someone give GEOMETRICAL interpretation of the meaning of limit and contuinity?
What does it mean to say that a ...
0
votes
0answers
142 views
Pre-requisites for the Calculus of Variations
I'm interested in working through the book : "Calculus of Variations" by Gelfand and Fomin. However, I lack the pre-requisites to do so (I'm familiar with linear algebra and one-variable calculus ...
3
votes
1answer
194 views
How might we find $\sigma$?
How does one solve a "differential equation" for $\sigma$ of the form
$$
\sigma(v)w_i(v)={\partial \over \partial v_j}\left[\sigma(v)A_{ij}(v)\right]
\quad i=1,\dots,n.
$$
where the summation ...
0
votes
1answer
62 views
Class of functions with negative mixed partial derivatives
Lets say we define a class of functions $g: \mathbf{R}^2 \rightarrow \mathbf{R}$ by the requirement that
$$ \frac{\partial^2 g}{\partial x_1 \partial x_2}(x_1,x_2) \le 0 $$
for all $x_1$ and $x_2$. ...
2
votes
0answers
132 views
Why is del operator coordinate free?
In solving Laplace equation $\Delta u=0$, every textbook will tell you to transform $\Delta=\partial_{xx}+\partial_{yy}$ into polar coordinate form $\Delta_p$(What it looks like doesn't matter here). ...
4
votes
2answers
199 views
how to understand the differential operator acting on functions that are not scalar
Quite often these days I find myself in a situation where I'd like to understand differential operators. One bit that is particularly subtle to me at the moment is how a differential operator is to be ...
1
vote
0answers
40 views
How we can show this?
Let $x \in \mathbb{R}^{3} $ and let $ f,g: \mathbb{R}^{3} \to \mathbb{R} $ be smooth functions. Define $F(x)= \nabla f\times \nabla g $ and let $r(t)$ satisfy the diffrential equation $ ...
1
vote
3answers
813 views
Particular solution to $y'' - 3y' + 2y = 2e^x$
Im trying to find the particular solution to: $$y'' - 3y' + 2y = 2e^x$$
I already have the homogenous solution so this is not my problem.
Assuming that $y_p = Ae^x \to {y'}_p = Ae^x ...
2
votes
0answers
135 views
Implicit function theorem and consistency of a semi-explicit DAE
This may be a trivial question, but here goes:
Suppose a semi-explicit differential-algebraic equation (DAE) system is defined as follows:
$$
\begin{align}
&\dot x = f(x,z,\theta),\qquad x(0) = ...
4
votes
2answers
3k views
Why is $dy dx = r dr d \theta$ [duplicate]
Possible Duplicate:
Explain $\iint \mathrm dx\mathrm dy = \iint r \mathrm d\alpha\mathrm dr$
I'm reading the proof of Gaussian integration. When we change to polar coordinates, why do we ...
2
votes
2answers
863 views
Solving the Nonlinear Second-Order ODE $u'' - u^2 + 9 = 0$
This is the one question from my prelim that still stumps me.
Consider the nonlinear ordinary differential equation
$$ u'' - u^2 + 9 = 0. $$
Convert the equation to a system of ...
2
votes
1answer
212 views
Setting up and solving differential equation with The Euler Method
I recently started this question and it gave me some insight into the world of differential equations.
However the solution was not fit for my goals as I wanted a general method for calculating the ...
1
vote
0answers
43 views
Need practical help with a calculation
I'm sorry for a really basic question. I lack proper background in mathematics, but I have to calculate a list of values.
I'm given a vector (list) of observations, and $\hat{Y}$, which is a list of ...
2
votes
1answer
166 views
Solving for a constant in a simple partial differential equation, using divergence/gradient
From a bank of past master's exams:
Let $u(x,y)$ satisfy $$ -\left( \frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}\right) = \lambda u $$ in a bounded region $\mathcal{D} ...
8
votes
2answers
615 views
What is the geometric interpretation behind the method of exact differential equations?
Given an equation in the form $M(x)dx + N(y)dy = 0$ we test that the partial derivative of $M$ with respect to $y$ is equal to the partial derivative of $N$ with respect to $x$. If they are equal, ...
