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

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2answers
22 views

What is the proper DE for those questions?

A tank starts with 500 liters of water with 1 kg of salt dissolved in it. A salt and water mixture with concentration 0.1 kg/L is poured into the tank at a rate of 2 L/min. The mixture is drained at 4 ...
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1answer
58 views

Assume that $f(t)$ is a known continuous function on $[0,\infty)$and $\lim_{t\to\infty} f(t)=2005$ [on hold]

Assume that $f(t)$ is a known continuous function on $[0,\infty)$and $\lim_{t\to\infty} f(t)=2005$ Consider a 1st order differential equation $dy/dt + 409y = f(t)$ a)Solve and write the general ...
2
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2answers
41 views

Differential of a rotated f(x, y) surface

I often hit this problem : Consider a surface defined by the equation $z = f(x, y)$, the differentials of this function are $\frac{\partial f}{\partial x}\mathrm{d}x$ and $\frac{\partial ...
2
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1answer
41 views

A counterexample for a smoth version of Tietze extension theorem

Is there any function $f:F\subset \mathbb{R}^2\rightarrow \mathbb{R}$ with $F$ closed such that $f|F$ is differentiable in every accumulation point but there is no differentiable extension to the ...
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1answer
24 views

ODE numerical solution [closed]

I tried it but ı cant create any code about this question.Does anybody help me.
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1answer
19 views

Find a particular solution that satisfies the initial condition [closed]

Find a particular solution that satisfies the initial condition $y y' - 2 e^x = 0$, $y(0)=3$
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0answers
39 views

How to solve first order second degree differential equation?

I'm trying to solve a differential equation, which, upon expanding gives a first order second degree differential equation. Here, $R$ is the radius of the Earth, $\mu$ is the frictional constant. Both ...
1
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1answer
18 views

Approximating monotonically increasing differential equation

I am trying to make sense of the Appendix of the paper (Cooper, 1986). The following model is presented: $$\dot{(BX)}=\gamma_1BX \\ \dot{(BXB)}=\gamma_2(BX)B \\ \dot{B}=\gamma_3(BXB)$$ Without ...
4
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4answers
110 views

What does $d\log\left(\frac{y}{x}\right)$ mean mathematically?

I am used to seeing derivatives written as $$\frac{df}{dx}.$$ But my economics professor keeps using notation like $$ d\log\left(\frac{y}{x}\right)$$ and I have no idea what this means. What does ...
0
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1answer
22 views

classify the critical point with given functions [closed]

The function $f(x, y)$ has a critical point where $f_{xx} = 2, f_{yy} = 1$ and $f_{xy} = f_{yx} = 1$. Classify the critical point.
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0answers
74 views

nonlinear integro-differential equation

I'm working on a engineering problem and I need to solve this nasty differential. I gave it a go with Laplace transforms, but no luck. Any ideas? Note: a, b, c, and k are constants. ...
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2answers
38 views

Is this writing incorrect?

If we want to find $\frac{d}{dx}\cos x^2 $ then is this writing incorrect $\frac{d}{dx} \cos x^2= \frac{d}{dx^2}\cos x^2 \times \frac{d}{dx} x^2 $
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1answer
13 views

The determinant of f is not invertible when f is zero when the norm of the function is constant.

Let $f:U\subset \mathbb{R^n}\rightarrow \mathbb{R}^n$ differentiable on the open $U$. If $|f(x)|$ is constant, then $Df(a)$ is not invertible for every $a\in U$. How can I prove that?
5
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1answer
70 views

If $ds$ is not a differential form, can I make sense of its intuitive notation somehow?

I understand that a line element is not actually a differential form but a $1$-density. My question is: is the notation $ds^2 = dx^2 + dy^2$ formal in any way? Can it be interpreted as outer or tensor ...
5
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2answers
91 views

How is an infinitesimal $dx$ different from $\Delta x\,$? [duplicate]

When I learned calc, I was always taught $$\frac{df}{dx}= f'(x) = \lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{(x+h)-x}$$ But I have heard $dx$ is called an infinitesimal and I don't know what this ...
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2answers
26 views

Find solution to the differential equation

$\frac{dB}{dx}+2B=50$ $B(1) = 50$ I tried separating the variables but that didn't work, and without separating the variable I'm not sure what to do.
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2answers
54 views

Find the solution to the differential equation

Assume $x\gt 0$ $$x(x+1)\frac{du}{dx} = u^2$$ $$u(1) = 4$$ I started off by doing some algebra to get: $$\frac{1}{u^2}du = \frac{1}{x^2+x}dx$$ I then took the partial fraction of the right side of ...
2
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2answers
20 views

Calculating the currvature of a tractrix

I'm trying to find the curvature of a tractrix expressed in the form $r(t)=(\sin{t},\cos{t}+ln(\tan{(\frac{t}{2}))} $. From what I've found on the Internet it appears that people arrive at the ...
1
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1answer
41 views

Curvature of plane curves

What is the neatest way to derive the formula for the curvature $\kappa =\frac{||y'x''-y''x'||}{(x'^2+y'^2)^{\frac{3}{2}}} $?
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0answers
17 views

Signed unit normal

I'm trying to study for one of my exams and the past papers have no solutions. I had to define the signed unit normal and the signed curvature. The signed curvature, $\kappa_s$ being such that ...
0
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2answers
63 views

Techniques to solve nonlinear first-order ODEs

I am trying to solve the following nonlinear ODE: $$\frac{dy}{dx} = \frac{1}{x(ayx-b)},$$ where $a, b$ are constants and $a>0$. Moreover, you may assume that $b \neq 0$ if necessary. This ...
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1answer
54 views

Is this equality correct?

I am working on a problem and stuck at some point. By intuition I believe that the equality below should hold. Then the bigger problem makes sense. However, I could not prove it. Does anybody prove or ...
0
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1answer
15 views

Second order differential with substitution

Hey guys, I was doing this question and am really stuck :/ I got up to taking n as 1 and getting z'=sqrt(y)*y' Can someone tell me where to go from here? Edit: I've done the first part, just not ...
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1answer
30 views

Function derivative question

My class is a bit late with the material, so we didn't have a lot of time studying function derivatives, so I am having a few problems with one of the questions I was given for practising for ...
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4answers
81 views

Euclidean norm second derivative

I really need Your help. I need to prove that Euclidean norm is strictly convex. I know that a function is strictly convex if $f ''(x)>0$. Can I use it for Euclidean norm and how? ...
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0answers
20 views

Twice differentiable functions that are harmonic

This is a question that I have spotted in a textbook for differential geometry. Determine all twice differentiable non-zero functions g : R $\rightarrow$ R and h : R $\rightarrow$ R such that $f ...
0
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1answer
19 views

Expression for $dW$ for a 3D position dependent force $\vec{F}(\vec{r})$.

I was looking at the derivation of the infinitesimal element of work done for a 3d position dependent force and I couldn't get over the switching of $\text{d}\vec{v}$ and $\text{d}\vec{r}$ in the ...
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2answers
21 views

Differential Equation Modeling

Quick disclaimer: This is not graded homework - all homework is assigned but not turned in. There is no assigned book, and hence no answers to given problems. These questions are for the purpose ...
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1answer
34 views

integral curves of vector fields

If we have a vector field on a boundary less and compact 2-manifold, which is neither a gradient nor a harmonic, does that imply its integral curves are closed?
5
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2answers
246 views

differentiate matrix exponential

I know this: $$\frac{d}{dt}e^{At} = Ae^{At}$$ However, in one lecture, I find the following: $$\frac{d}{dt}e^{A^Tt} = e^{A^Tt}A^T$$ The lecture is as following: How to show the second case? ...
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1answer
38 views

Series Solutions Near an Ordinary Point 5.2 #3

Question 3 of 5.2 in Boyce's Differential Calculus asks a) Seek power series solutions around x_0 and find the recurrence relation b) Find the first four terms in y1 and y2 c) Show that y1 and y2 ...
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2answers
122 views

Differential Forms Notation is Wrong? Confirm or deny? [closed]

Being an engineering student that just happens to have a large interest in math, I have always felt that appealing to definitions instead of intuitively understanding a concept is a mistake. A while ...
3
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0answers
58 views

Solving a differential equation with a square root

I am trying to solve the differential equation $ A(x)\frac{d^{2}f(x)}{dx^{2}}+B(x)\frac{df(x)}{dx}=\frac{1}{3}\frac{1}{\sqrt{f(x)}}, $ where $ A(x)=\frac{x}{x+1} $ and $ B(x)=\frac{2x+1}{(x+1)^{2}} ...
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1answer
27 views

Solving $y'-\frac y x=0$ with integrating factor

How do you solve $y'-\frac y x=0$ the answer should be $y=ex$ but I can't get to that point. I tried using the integrating factor but I can't get it to add up.
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1answer
18 views

Differential Equation Modeling high-school

I've encountered a problem I cannot seem to be able to solve. 1 = the problem 2 = my solution _____1 A ball has the volume of 3.0 cm^3. The volume decreases with time t (in months), the change per ...
2
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0answers
38 views

Find the curve. Differential Geometry.

Find a non plane, closed curve such that the plane curve with the same curvature as function of the arclength is not closed. Been thinking a lot in this problem and haven't got a clue. ¿Any ...
4
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0answers
67 views

Why is the wedge product associative?

I have been reading on the wedge product (From Shutz's Geometrical Methods of Mathematical Physics) and I don't quite get why the wedge product is associative. The book defines the wedge product of ...
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0answers
31 views

formulate a differential equation that expresses how N is changing with respect to time.

A country has 12 billion dollars in paper currency in circulation. The government wants to introduce new currency by having banks replace old bills with new bills whenever old currency comes into the ...
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0answers
59 views

What is a good conceptual interpretation of a differential?

I'm having trouble with understanding what exactly a differential really is. For example, if we have the following function, $f(x,y)=x^2+xy+\frac{37}{x} +5$, does this differential, ...
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1answer
36 views

How to proceed for these two differential equations?

1) Using $z=x+y$, solve $$\frac{dy}{dx}=\frac{x+y+2}{x+y+5}$$ My attempt,so $$\frac{dy}{dx}=\frac{z+2}{z+5}$$ How to integrate then to become y? 2)Using $v=2x-y$, solve ...
2
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1answer
48 views

Bernstein's Theorem of Analytic Function Proof

I'm studying from a textbook and came across an exercise to prove the following, which it calls the Bernstein's Theorem: If $f$ is infinitely differentiable on an interval $I$, and $f^n(x)\ge0$ for ...
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1answer
33 views

Are both interpretations of the differential accurate?

A differential can be written, informally, as the form of $du=\dfrac{\partial f}{\partial x_1}dx_1+\dfrac{\partial f}{\partial x_2}dx_2+...\dfrac{\partial f}{\partial x_n}dx_n$. In the textbook I am ...
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1answer
38 views

Differential calculus

A ladder 20 feet long leans against a vertical building.If the bottom of the ladder slides away from the building horizontally at a rate of 3ft/sec,how fast is the ladder sliding down the building ...
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2answers
20 views

DE undetermined coefficients

De problem $$x'' + 196x = 40\sin (14t)$$ I have the general solution, I'm just struggling with the algebra with the undetermined coefficient. I set my guess as: $At\cos (14t) + Bt\sin (14t)$, took ...
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0answers
28 views

is a function differentiable at the point

$G(x,y)=(x^2+y,xy+x^2)$, $P=(a,b)$ is G differentiable at P? calculate dG(P) Attempt: I think I understand how to find dG(P) it is just, $dG(P) = (2a,b)$ correct? I am needing help how I show that ...
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2answers
79 views

Undamped spring mass system

I have this study guide for an upcoming test for DE class I'm trying to figure out. A mass of 400 grams stretches a spring by 5 centimeters. (a) Find the spring constant k, the angular frequency ω, ...
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1answer
12 views

Expressing following in one equation

I have 3 variables: theta (optimal launch angle), h (launch height), v0 (velocity at launch). h and v0 are independent from each other. How do I express theta in terms of h and v0, if I have ...
1
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2answers
54 views

Using differentials to optimize a function

I've read in a paper by Tevian Dray an alternative way to solve optimization problems manipulating "differentials". Here is an example of how it works (next I quote the paper). Consider the ...
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0answers
17 views

Laplacian Equation in general

Say, I have a function $\phi$ of two variables ($x$ and $y$) My objective is not to solve this equation exactly but to look for the order of terms. I know that the scalar $\phi$ has following ...
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0answers
19 views

How to solve: $ e^{-u} \frac{d^2u}{dy^2}= -\delta e^{-2y} $

How can I approach this non-linear ODE $$ e^{-u} \frac{d^2u}{dy^2}= \delta e^{-2y} $$ which I derived on this question? Any ideas? Can we consider this a separable ODE, eventhough it cannot be ...