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Questions tagged [differential]

For question about the differential of a map from an open set of a vector space to a vector space.

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Eigenvalues of differential equation are sines of integers times a constant

We know of differential equations whose eigenvalues are integers "m". Can anyone point to a differential equation whose eigenvalues are terms of the form $\sin(ma)/\sin(a)$, which reduces to integers ...
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Differential geometry and frenet formula

Given a curve C by its arclenght (vector $r(s)$), prove that $\frac{dT(s)}{ds} \times \frac{d^2 T(s)}{ds^2} = k^2 \omega$ where k is the curvature and $\omega$ is the darboux vector. I tried using ...
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1answer
23 views

Use Duhamel's principle in PDE

Let $u(t,x)$ be solution for: $$ u_{tt}-u_{xx}=q(t,x)u$$ with zero initial conditions. If $q$ is bounded in $\mathbb{R}^2$, show that $u \equiv 0$ using Duhamel's principle
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Integrating Factor Techniques for Exact ODE

As explained in this answer, an inexact ODE in the form $M(x,\ y)\ dx + N(x,\ y)\ dy = 0$ can be transformed into an exact ODE via multiple integrating factors which vary from each other non-trivially ...
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Calculate a gradient

Given a constant $c\in\mathbb{R}^n$, and given $n+1$ constant vectors $u_0,...,u_n\in\mathbb{R}^n$ such that $u_1-u_0,...,u_n-u_0$ form a basis of $\mathbb{R}^n$ (ie they are affine-independant), let $...
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maximum error and percentage relative error in surface area.

The radius and altitude of a closed (has a top and bottom) right circular cylinder is measured as 5 inches and 9 inches respectively. There is a possible measurement error in radius of ...
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1answer
31 views

Two bodies under mutual gravitational attraction as system of first order ODEs

I'm trying to model the motion of one mass $m$ about another larger mass $M$. I have the following relation: \begin{align} \ddot x_0 &= -GMx_0/r^3, \\ \ddot x_1 &= -GMx_1/r^3, \end{align} ...
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1answer
33 views

An IVP with singularity having continuous solutions

I am TAing undergraduate differential equation course, and recently have encountered the following problem: Solve the following initial value problem $$ y' + (\tan x) \cdot y = \cos ^2 x, \quad y(0) ...
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The solution formula for a homogeneous linear differential equation of order n with constant coefficients

The solution formula for a homogeneous linear differential equation of order n with constant coefficients? I have trouble finding one myself. I wish for it to be as ”easy” (decide yourself what easy ...
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1answer
19 views

Exact equations, First order differential equations

If xM (x, y)+yN (x,y) = 0, where x, y, M (x, y), and N (x, y) are non-zero, solve the differential equation M (x, y) dx + N (x, y) dy = 0:
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How to solve this differential equation $\frac{r^2\partial_r^2R(r)}{R(r)}=(l+l)l$?

How to solve this differential equation $\frac{r^2\partial_r^2R(r)}{R(r)}=(l+l)l$ where $l$ was some positive integer.
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Differential Mixing Problem

At time $t=0$, a tank contains 30 oz of salt dissolved in 100 gallons of water. Then brine containing 8oz of salt per gallon of brine is allowed to enter the tank at a rate of 2 gal/min and the mixed ...
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1answer
127 views

Exact Differential Equation Geometry

In a variety of contexts, I have noticed hints of a strong connection between exact differential equations and machinery from multivariable calculus. From another question, I have gathered that the ...
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28 views

PDE-find general solution and solve initial value problem [closed]

I stuck with method of characteristics here, how to find the general solution and IVP? ${u_{xx} + 4u_{xy}+3u_{yy}}={0}, -\infty <x<+\infty, t >0 $ $ u(x,0)=0, -\infty <x<+\infty, $ ...
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a system of simultaneous linear differential equations with constant coefficients

Solve the system of differential equations: $$ \frac{\mathrm{d}}{\mathrm{d}t}x=2y, \quad \frac{\mathrm{d}}{\mathrm{d}t}y=2x, \quad \frac{\mathrm{d}}{\mathrm{d}t}z=2x $$ I tried solving the first ...
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2answers
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PDE- method of characteristics, satisfy the given condition

How to solve this equation? Should I use method of characteristics? Question states: find the solution that satisfies this condition: \begin{aligned} xu_{x}-yu_y+u &= x\\ u&= x^2 \ when \ y=x ...
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Prove that there exist $x_1,x_2,…,x_{100}\in(1918,2018)$ such that $\sum_{k=1}^{100}f'(x_k)=100$

Let $f:[1918,2018]\rightarrow\mathbb{R}$, continuous over $[1918,2018]$, differentiable on$(1918,2018)$ and such that $f(1918)=1918,f(2018)=2018$. $a)$Prove that there exist $x_1,x_2,...,x_{100}\in(...
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1answer
76 views

$\frac{dy}{dx} - e^{x-y} + e^y = 0 $ [closed]

how can I solve the given first order differential equation: $ \frac{dy}{dx} - e^{x-y} + e^y = 0 $
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1answer
29 views

Derivative with respect to vectorized inverse Kronecker product

I am trying to derive the gradient of a function I wish to optimize, and wish to obtain the following derivative: $$ \frac{\partial}{\partial \pmb{x}} \left(\pmb{I} - \pmb{X} \otimes \pmb{X} \right)^{-...
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1answer
25 views

Kinematics of a disc

So the kinematics of the contact point of a disc rolling without slip in a Cartesian plane if fairly straightforward. The velocity for the contact point is just v = ω r ev where r is the wheel radius ...
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1answer
42 views

Topology on the n-dimensional sphere using stereographic projection?

Let the n-dimensional sphere be defined as the set: $$ \mathbb{S}^{n-1} := \{(x_1, x_2,..., x_n)\in \mathbb{R}^{n-1}: x_1^2 + x_2^2+...+x_n^2 = 1 \}$$ If we cover $\mathbb{S}^{n-1}$ with this Atlas ...
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41 views

Total differential equation with an integrating factor depending on the product $xy$

Description Show that if the quantity $$\frac{\frac{\partial P(x,y)}{\partial y}-\frac{\partial Q(x,y)}{\partial x}}{yQ(x,y)-xP(x,y)}$$ is a function $g(z)$ of the product $z=xy$, then the quantity: ...
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1answer
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General solution using Integration

I have a DE like this, $$X'' - 4a^2X = 0$$ and its solution is $$X = c_1 \cosh 2ax + c_2 \sinh 2ax$$ Now, I want to know in detail how would I get this solution. I need all the steps to get the ...
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Differentials practice question

For $f(x1,x2) = x_2cos(x_1)$ find $f'(x_1,x_2)$, $f'(2,7)$ and $df[(2,7);h].$ I got $f'(x_1,x_2)=[-x_2sin(x_1) ,cos(x_1)]$ and $f'(2,7)=[-7sin(2), cos(2)]$ Are those correct? And how would I get $df[...
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1answer
21 views

Differential Equation with arctanx

I have a separable equation $\frac{dy}{dx}$=$\frac{27}{y^{1/3}+81x^2y^{1/3}}$ I separated both sides by multiplying by dx and factoring out the y^{1/3} and ...
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1answer
55 views

Equations of Motion in Cylindrical Co-ordinates

I've run into an interesting set of differential equations, that I'm not 100% sure where to begin- I'm not looking for a 100% complete solution, more just a push in the right direction of where I can ...
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1answer
32 views

Initial Value Problem with continuous functions

Hello I haven't seen a question like this before and would appreciate any help or guidance with the question: Let $x, y : I \to \Bbb R^N$ , where $N \in \Bbb N$, be solutions of $$x'(t) = f(t)$$ $$...
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In which frame are $d$ and $\partial$ rigourous?

I know $\frac{dy}{dx}$ may be a shortcut of $\lim_{h \to 0} \frac{y(x+h)-y(x)}{h}$, which is totally rigourous, but it loses that sense if I write $dx=f\cdot dy$. How could that become rigourous ...
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1answer
59 views

Solution to a differential equation involving inseparable variables.

What is the solution for the following DE $\frac{dy}{dx} - \epsilon{y} = x$ Where $0 \leq x \leq 1$ and initial condition y = 1 when x = 0 and where $\epsilon$ is any positive parameter
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2answers
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Prove that function is continuous for all x ∈ R / {1}. f(x)=$\frac{1}{x+1}$

I have to use a definition of continuity of function ( if function is continuous at point x then Δf(x) tends to 0 as Δx tends to 0) to prove than function is continuous. This is the function: $$\...
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Finding a Linear Second Order Differential Equation $F(x, y, y', y'') = 0$ for which $y = c_1x + c_2x^3$ [closed]

Find a linear second-order differential equation $F(x, y, y', y'') = 0$ for which $y = c_1x + c_2x^3$ is a two-parameter family of solutions. Make sure that your equation is free of the arbitrary ...
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A solution for a PDE

Is there somebody who can help me to find a solution for $$\begin{cases} y^i\frac{\partial b(x,y)}{\partial x^i}=0\\ y^i\frac{\partial b(x,y)}{\partial y^i}=b\\ \end{cases}, b:R^{2n}→R, x=(x^1,…,x^n)\...
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1answer
21 views

Existence of solutions for an Ordinary differential equation modelling a real physical problem [closed]

Does an ODE (or Initial value problem) modelling a real physical problem always have a solution, whatever the initial conditions? If so, can this be extended to Partial differential equations as well?
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51 views

Differentials and Integration

I have been informed that consecutive differentials in iterated integral problems are actually connected via the exterior product. So the factor $dx\ dy$ in $\int\int x^2\ dx\ dy$ is actually the ...
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1answer
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Lipschitz Continous Function and Differentiable. Does this imply it derivative is bounded?

While studying differential equations, I came across the existence and uniqueness of solutions for a differential equation. Then after studying some theorems these statements are troubling me. Let $f:...
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45 views

condition for one-form to be exact differential

Simple question I suppose: Having a (smooth) one-form $$\omega=\omega_i dx^i\in \Lambda(M)$$ Is there a test to find out if $\omega$ is exact differential? Or more precisely that there exists a ...
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Computing the differential of a Lie group action

(Ex. 27.4 page 252 Loring Tu) (The differential of an action). Let $\mu: P \times G \rightarrow P$. For $g \in G$, the tangent space $T_gG$ may be identified with $l_{g*} \mathfrak{g} $, where $...
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Differential Equation help me solve

Help me solve this Differential Equation? $$y\cdot y''+(y')^2=2x$$
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Finding a curve orthogonally equidistant between two curves.

I have two curves, $f_1(x)$ and $f_2(x)$ which meet at $(x_0, 0)$ and $(x_1, 0)$, these curves as such form a closed shape, but not necessarily a convex one. I would like to generate a curve $f_c(x)$ ...
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derivative exist almost everywhere

I am working on a problem finding an error of an example that uses integration by parts. The condition of the example says that $f'$ and $g'$ exist almost everywhere. If $f'$ and $g'$ exist almost ...
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SVM “kernel trick” and linearly separable sets of points

Background info: in Machine Learning there's something called an SVM (Support Vector Machine) that employs a "kernel trick" to map sets of points to a higher dimension in order to find a hyperplane ...
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Find a gradient system admitting a given solution

Consider the Gradient Dynamical System, \begin{align*} \begin{cases} \dot{\gamma}(t)=-\nabla f(\gamma(t)),t>0,\\ \gamma(0)=x_0\in\mathbb{R}^n. \end{cases} \end{align*} We all know that if $f$ is $C^...
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Algebraic Groups, dual numbers and differentials

I was looking for a method to compute the explicit differential of a regular map between algebraic groups. More precisely if $X$ is a sub-variety in an algebraic group $G$ (say over a finite field $k$)...
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Bernoulli differential equation alike

I am quite new to differential equations and I have the following $$ \partial u(x)/\partial x = a(x)u(x)^2+b(x)u(x)+c(x) $$ which is, to the best of my knowledge, not exactly a Bernoulli differential ...
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Is this function continuous and differentiable?

The function is $G(x,y)= 1$ if $y \neq e^{x}$ and $0$ if $y= e^{x}$.
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1answer
31 views

Find the product of maximum and minimum value of $f(x)=\sqrt{2}\sin{x}-f(0)$

$f(x)=\sqrt{2}\sin{x}-f(0)$ (A) $\frac{-3\sqrt{2}}{4}$ (B) $\frac{-3\sqrt{2}}{2}$ (C) $\frac{-3}{2}$ (D) $\frac{-3}{4}$ (E) $\frac{-7}{4}$ My attempt : let x=0 $f(0)=\sqrt{2}\sin{0}-f(0)\\ f(0)...
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Rewrite differential equation [closed]

If $h(t+1) = g \cdot h(t) - dh(t)/dx$, then how can you rephrase it to $dh/dt = \dots$ ? It's an advection equation describing the evolution of a fluid h over time. Thanks
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218 views

Solving $(\ln(x)-1)y'' - \frac{1}{x}y' + \frac{1}{x^2}y = \frac{(\ln(x) - 1)^2}{x}$

On my exam I had to solve the following differential equation. \begin{equation} (\ln(x)-1)y'' - \frac{1}{x}y' + \frac{1}{x^2}y = \frac{(\ln(x) - 1)^2}{x^2} \end{equation} Which is a differential ...
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Derivative of sum of Gaussians wrt. sigma - How does this work?

Let $\varepsilon_i \sim \mathcal{N}(0,\sigma_i^2)$ be samples from independent Gaussians, we know by additivity of independent Gaussians that $\varepsilon_1 + \varepsilon_2 = \varepsilon'$ where $\...