0
votes
1answer
6 views

Solving $\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = 0$ by changing variables

Transform the differential equation $\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = 0 $ by introducing new variables $x = u+v$ and $y=u-v$. then solve it. I which I could show ...
1
vote
1answer
28 views

Particular solution of reducible to homogeneous equation

Verify that $y=x-5$ is a particular solution of the equation $$\frac{dy}{dx} = \frac{2y+6}{x+y+1}\ .$$ This is when $y'=1$ but this is not given as a condition in the question. How would you write the ...
1
vote
0answers
65 views

Computation of the Frenet-Serret trihedron in $\Bbb L^3$ (Lorentz-Minkowski space)

Consider $\Bbb L^3 = (\Bbb R^3, \langle , \rangle)$, with the convention $$\langle (x_1,y_1,z_1), (x_2,y_2,z_2)\rangle = x_1x_2+y_1y_2 - z_1z_2$$ and $\| v \| = \sqrt{|\langle v, v \rangle|}$. Let ...
3
votes
2answers
35 views

Variable substitution in second order PDE

Consider the the PDE $$A(x, y)\partial_{xx} u + B(x, y)\partial_{xy}u + C(x, y)\partial_{yy}u=h(x, y) $$ Now I want to make a variable substitution $\xi=f(x, y), \eta=g(x,y)$, so I can get $u$ as a ...
2
votes
1answer
36 views

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m $ be differentiable and $K \subset \mathbb{R}^n$ be compact and convex. Show $f$ is Lipschitz on $K$.

The Assignment: Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m $ be continuously partial differentiable and let $K \subset \mathbb{R}^n$ be compact and convex. Show $f$ is Lipschitz on $K$. A ...
0
votes
3answers
117 views

Is speed a function of position?

Let $x$ be a smooth function from $[0,\infty)$ to $\mathbb{R}^n$ satisfying the following differential equation $x''(t) = f(x(t))$, where $f$ is a smooth function from $\mathbb{R}^n$ to itself. Then ...
0
votes
1answer
23 views

Let U be open and $f: U \rightarrow \mathbb{R}$ be partial differentiable.

The Assignment: Let $U \subset \mathbb{R}^n$ be open and $f : U \rightarrow \mathbb{R}$ be partial differentiable and let all partial directional derivatives be continous function on $U$. Show ...
1
vote
0answers
17 views

Partial differentiation for multi variables

A Candy company makes 2 types of candy A & B for which the average costs are 2 & 3euros per kg respectively. $Q_a$ & $Q_b$ (a & b subscript) are the kg that can be sold each week and ...
5
votes
0answers
168 views

The Stable Manifold Theorem Applications

Definition: Let $\phi_t(x)$ be the flow of the nonlinear system $x'=f(x)$. The global stable manifold of $x'=f(x)$ at $0$ is defined by: $$W^s(0)=\bigcup_{t\leq 0}\phi_t(S)$$ Where $S$ is a ...
0
votes
0answers
46 views

Calculus based question

For those who knows this stuff this is probably an easy question, but I don't know in general where I should consult to learn solution of this type of problems. So, beside providing solution/hint if ...
2
votes
5answers
102 views

How to think when solving $3\frac{\partial f}{\partial x}+5\frac{\partial f}{\partial y}=0$?

Solve this differential equation $$3\frac{\partial f}{\partial x}+5\frac{\partial f}{\partial y}=0$$ Usually, when we get these problems, they tell us what variable change is smart to do and we just ...
0
votes
1answer
21 views

existence of the solution of Neumann problem in $\mathbb{R}^3$

Let $D\subset \mathbb{R}^3$. Let $D$ be connected subset of $\mathbb{R}^3$. Show that there is not any solution of the system of equation \begin{equation} \Delta u=f \text{ in } D, ...
0
votes
0answers
21 views

Green's first or second or both identities?

(i) Let $F(x) = (-1/2) e^{-\mid x\mid}$. Show that $0=\int_{\mid x \mid >\epsilon} (F''-F) \varphi =\int_{\mid x \mid >\epsilon} (\varphi'' -\varphi)F + \int_{\mid x \mid >\epsilon}F''\varphi ...
1
vote
1answer
29 views

Chain rule and multivariable differentiation

Given that $F: (\mathbb{R}^3) \longrightarrow (\mathbb{R}^2)$ is defined as $F(x,y,z) = (x^2+y+z, x+y^2+xz)$, find all the linear functions defined as $g: (\mathbb{R}^2) \longrightarrow ...
1
vote
0answers
54 views

How to compute time ordered Exponential?

So say you have a matrix dependent on a variable t: $$A(t)$$ how do you compute $$e^{A(t)}$$ It seems Sylvester's formula, my standard method of computing matrix exponentials can't be applied ...
2
votes
1answer
126 views

How do I solve $F = \nabla\times G$ for $G$?

Given the vector valued function $F(x,y,z) = (xz,-yz,y)$ find $G$ such that $F = \nabla\times G$ I let $G(x,y,z) = (G_1,G_2,G_3)$ and expanded $\nabla \times G$ then equated the components to $F$ but ...
2
votes
1answer
31 views

Solution in common for two differential equations

Consider: $E1: y''-4y'+4y=0$ Solution: $y(x)=c_1 e^{2x}+c_2 x e^{2x} $ $E2: y''-2ay'+(a^2-1)y=0$ Solution: $y(x)=c_1 e^{(a+1)x}+c_2 e^{(a-1)x} $ For what values of $a$, $E1$ and $E2$ have ...
0
votes
0answers
57 views

Derivation of Euler Lagrange Equation

I was reading on the derivation of the Euler Lagrange Equations (in the link: http://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation focusing on: "Derivation of one-dimensional Euler–Lagrange ...
1
vote
1answer
44 views

Need helping proving that something is differentiable but not continuously differentiable

I need some help please proving that a function is differentiable at $(0,0)$ but not continuously differentiable at $(0,0)$. The function is as follows... (from $\mathbb{R}^2$ to $\mathbb{R}$) ...
0
votes
0answers
27 views

predictor-corrector method and stability

A predictor-corrector method for the approximate solution of $y'=f(t,y)$ uses \begin{equation} y_{n+1}-y_{n}=hf_{n} \tag P \end{equation} as predictor and \begin{equation} ...
0
votes
1answer
25 views

Solution for a differential equation

If $x^n$ is a solution for the equation: $y''y'y=48x^9$, what is the value of $n$? I have the choices: 2, 1, 3, 4, or 6. I dont know how to resolve $y''y'y=0$, but maybe i dont need to do it. ...
0
votes
2answers
28 views

Find coeficients for differential equation given a solution

For: $y''+ay'+by=0$ Given one of the solutions: $y(x)=e^x cos(x)$ Whats the value of $a+b$ ? I need some help with this, i dont know where to start.
0
votes
1answer
41 views

Differential Equations - Method of Undetermined Coefficients for products of polynomials and sines

Consider $y''+y= 2x \sin (x)$ I have the solution for the homogeneous equation. Now i am trying to guess a particular solution for: $2x \sin (x)$ My first guess was: $(Ax+B) \cos x + (Cx +D) \sin ...
1
vote
0answers
49 views

Linearization of Implicit ODE (Equations of Motion)

let's say we have a system with vector $q_{(t)}$ representing the degrees of freedom (DoF), and state vector $ x_{(t)} = \left \{ \begin{array}{c c} q_{(t)} \\ \dot{q_{(t)}} \end{array} \right \}$ ...
0
votes
1answer
27 views

how to introduce time into calculus of variations for image processing?

I'm studying some topics about calculus of variation applied to image processing. I'd like to understand how to introduce time parameter to evolve an image in an iterative way. For example, let's ...
1
vote
2answers
37 views

Differential equation have no particular solution for initial condition

Does the following diferential equation have a particular solution? $y'=(y-1)(y+1), y(0)=1$ The general solution is: $\frac{1-e^{2x+2C}}{1+e^{2x+2C}}$ But then: $\frac{1-e^{2C}}{1+e^{2C}}=1$ $C$ ...
1
vote
2answers
46 views

How to resolve this diferential equation $y^2 y^{\prime}=x^3$

I see it is non-linear, but not sure if that is important here. I got the solution for the homogeneous in this way: $$y^2 y^\prime=0 \rightarrow y^\prime=0 \rightarrow \frac{dy}{dx}=0\rightarrow ...
0
votes
2answers
74 views

Minimizing the following objective function with matrices

Suppose $A$ and $B$ are known matrices, and we are to find matrix $X$ that minimizes the following function, $$\frac{1}{2}||X||^2+\frac{1}{2}||X^TAX-B||^2$$ Taking the relevant derivative w.r.t $X$ ...
0
votes
1answer
44 views

Compatibility Condition of the Poisson Equation with Neumann Boundary Conditions

I am trying to solve the following general Poisson equation with homogeneous Neumann boundary conditions in a rectangular domain ($0 \le x \le L$ and $0 \le y \le H$). $$ \frac{\partial^2 ...
0
votes
2answers
38 views

Finding function given its Jacobian and the initial condition

Consider continuously differentiable function $f:\mathbb{R}^k\mapsto \mathbb{R}^k$. We know that $f(x_0)=y_0$ and the Jacobian matrix is given for all $x$. I'd like to know the explicit for of the ...
0
votes
1answer
39 views

$\textbf c(t)$ is a flow line on $\textbf F = -\nabla V$, prove $V(\textbf c(t))$ is a decreasing function of $t$.

Let $\textbf c(t)$ be a flow line of a gradient field $\textbf F = -\nabla V$. Prove that $V(\textbf c(t))$ is a decreasing function of $t$. We have not learned Line Integrals, so I would assume this ...
0
votes
1answer
42 views

how to solve this homogeneous differention equation ?

$dy/dx= (2x+3y+4)/(4x+6y+5)$. I am trying to solve this homogeneous Ds, but don't understand how to solve it. I believe the first step is to solve this: 1) $y=u x$ 2) $dy/dx = u+x \cdot du/dx$ then ...
5
votes
2answers
300 views

A proof of a theorem of Liouville

I need some reference for the proof of the following theorem attributed to Liouville: Theorem: Let $f(x):\Omega\longrightarrow \mathbb R^n$ a $C^2$ function where $\Omega$ is an open subset of ...
-1
votes
0answers
29 views

derivation of higher order numerical methods for ODEs using Mathematica, Matlab, Maple

I want to know that is it possible to expand the multi-variable Taylor series of f(x+ah,y+ bhf(x+ch,y+dh)) in Mathematica. My purpose is to construct higher order methods which is very typical to ...
1
vote
1answer
55 views

Multivariable differential equation

Given $u=f(2x-y)+g(x-2y)$, show that $$2 \frac{\mathrm{d}^2 u}{\mathrm{d}x^2} + 5 \frac{\mathrm{d}^2 u}{\mathrm{d}x\,\mathrm{d}y} + 2\frac{\mathrm{d}^2 u}{\mathrm{d}y^2} = 0.$$ I'm not even sure where ...
2
votes
1answer
48 views

Second Order Derivative of a function $f:R^2\to R^2$

The Exercise: My Work: Part 1: $$ Df=\left( \begin{array}{ccc} D_1f_1 & D_2f_1\\ D_1f_2 & D_2f_2 \\\end{array} \right) $$ $$f_1(x,y)=\sin x+\sin y$$ $$f_2(x,y)=\cos x+\cos y$$ $$ ...
0
votes
0answers
30 views

Existence and Uniqueness Theorem--Question Regarding Regions

I have a question. Suppose I have a differential equation for which I want to find the values at which $f(x,y)$ and $∂f/∂y$ are discontinuous, that I might know the points at which more than one ...
1
vote
0answers
47 views

Directional derivatives, linear maps, and uniform convergence

The Exercise Let $f(x,y)=x$ if $|y|>x^2$ and $f(x,y)=0$ otherwise. Show that all the directional derivatives of $f$ exist at the origin but there does not exist a linear map $D$ such that ...
0
votes
1answer
25 views

Directional Differentiability follows from Multivariate Differentiability

The Exercise: "Suppose that a function $f: R^n \to R^m$ is differentiable at $x \in R^n$. Show that the directional derivative of $f$ in any direction $v\in R^n$ at $x$ exists and $D_vf(x)=Df(x)(v)$" ...
2
votes
1answer
86 views

Derivative of a Linear Map

I'm devastatingly incompetent at linear algebra and multivariable calculus. I just cannot understand it at all. Here's the easiest problem from my homework, and my attempt at solving it, and where I ...
0
votes
1answer
82 views

Integrating two equations that equal, what happens to the constant on one of the sides?

In class, we were talking about Newton's 3rd law and how to integrate. $\int(g)dt = \int(y''(t))dt \implies g(t) + C = y'(t)$ I am confused about why the right hand side of the equation doesn't get ...
2
votes
2answers
103 views

Tricky Differential Equation Problem

I am unsure of how to tackle the following differential equation: $$ dx+ x\,dy = e^{-y}\sec^2y\,dy$$ I have done the following so far: $$dx + x\,dy = e^{-y} \sec^2y \, dy$$ $$=>dx = e^{-y} ...
0
votes
0answers
19 views

Could anyone help me differentiate an ansatz for the Calogero-Marchioro model?

Could anyone help me substitute the following ansatz, $$\psi=(\prod_{i<j}^{N}\bf{r_{ij}^{2}})^{\frac{\Lambda_D}{2}}\phi(p)$$ into $$H=-\frac{{\hbar}^{2}}{2m}\sum_{i=1}^{N}{\nabla}_i^{2}$$ So ...
2
votes
1answer
59 views

Four equations with three variables!

I have these four equations: $$\left\{\begin{matrix} A_1 = A_c -D \theta cos(\phi)\\ A_2 = A_c -D \theta cos(\phi-\pi/2)\\ A_3 = A_c -D \theta cos(\phi-\pi)\\ A_4 = A_c -D \theta cos(\phi-3\pi/2) ...
0
votes
1answer
66 views

Help with solving for a flow curve:

So I'm preparing for a final exam in multivariable and our textbook posed the following question: find the flow lines of F(x,y) = (-y, x) Which I can't seem to solve correctly. We are told that a ...
1
vote
0answers
72 views

Motion in three dimensions with friction.

I am trying to represent the motion of an object in three-dimensional space that is undergoing acceleration, friction, and drag, where: acceleration = $\vec A$ friction = $F$ drag = $D$ The ...
2
votes
2answers
63 views

How to find other solutions to this vectorproblem?

Suppose I have a vector field $\mathbf{A}(x,y,z)$, of which I know: $$ \mathbf{A}(x,y,0)=(1+\alpha x)\hat{z}$$ Thus, I know the value of $\mathbf{A}$ in the $xy$-plane. Say, within ...
0
votes
0answers
88 views

analytical solution for system of nonlinear ODEs in two dimensions

This ODE comes out of a system of self propelled particles (http://www.foelsche.com/swarm). Currently this system is linear -- I want to add a nonlinear friction/drag. Every particle has the following ...
1
vote
0answers
28 views

$o(|X(t+\Delta t)-X(t)|) $ is $o(\Delta t)$?

X is differentiable. if $X'(t) \neq 0 $ , it's easy to show. $ \lim_{\Delta t \to 0} \frac{o(|X(t+\Delta t)-X(t)|)}{|X(t+\Delta t)-X(t)|} \frac{|X(t+\Delta t)-X(t)|}{\Delta t} = 0 \cdot |X'(t)|$ ...
0
votes
0answers
30 views

arbitrary patch in terms of orthogonal patch

Let $\mathcal{M}$ be a $2$ dimensional differentiable manifold (a geometric surface). Let $\mathbf{y}$ be an arbitrary patch in $\mathcal{M}$. How can one prove that $y$ can be expressed as ...