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

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1answer
24 views

how to write a differential equation for a problem like this

I've got a problem and i should solve it using differential equation.I don't know how to write the equation and start. A person is trying to fill a bathtub with water. Water is flowing into the ...
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1answer
27 views

Compound interest Differential Equation

A college student starts a savings account with an initial balance of $\$0$. He plans to save money at a continuous rate of $\$200$ per week. Also, at every week he plans to increase this rate by ...
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0answers
46 views

Combining two differential equations

I have two differential equations that are connected by an equation, $L_1\frac{d^2I_1}{dt^2} + \frac{1}{C_1}I_1=\frac{dV}{dt}$ $L_2\frac{d^2I_2}{dt^2} + \frac{1}{C_2}I_2=\frac{dV}{dt}$ $I_1+I_2=I$ ...
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0answers
27 views

Solving Black scholes PDE using Laplace transform

I'm trying to obtain the Laplace transform of Call option price with repect to time to maturity under the CEV process. The well known Black scholes PDE is given by $$ ...
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0answers
21 views

State space equation of a RLC circuit

I have been trying this problem for last 4 hours and feeling stupid. I thought it was a easy problem, but I just can't figure out the state space equation. It should have three states (for three ...
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1answer
64 views

Interpretation of differential form

We know what is the interpretation of a total differential, ex.: $$df=\frac{\partial f}{\partial x} dx+ \frac{\partial f}{\partial y} dy$$ But what is the interpretation of a 1-form and its exterior ...
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0answers
12 views

find of value Zx and Zy

I try to solve this Z^2+ZY+XY=0, and find the value of ZX and ZY, but couldn't find any hint in the web neither in a couple of calculus books for this particular equation. I have no clue about how to ...
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2answers
73 views

Question about integrating differential forms

Maybe it's stupid question, by why: $$\int_S Fdx\wedge dy=\int_S Fdxdy$$ And is calculating a surface integral $$\int_S Fdx\wedge dy+Gdy \wedge dz+H dz\wedge dx=\int_S Fdxdy+\int_SGdydz + ...
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2answers
16 views

Differential Equation Word Problem involving y=Ce^(xk) (y=y')

"The rate of change of y is proportional to y. Write and solve the differential equation that models the verbal statement." This part of the problem is easy. My work is such: $y'=ky$ ...
2
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1answer
19 views

solving Legendre equation using the Frobenius method around a singular point

My first question is whether I can solve the Legendre equation for $l=0$, i.e. $$ (1-x^2)y''(x)-2xy'(x)+y(x)=0, $$ using the regular power series method around $x=0$. My second question is, even if ...
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0answers
5 views

Characterizing subdifferential of nuclear norm of X^T X!

I am interested characterizing the subdifferential of function f=|X^TX|_*. Basically its the nuclear norm of X^T X. Two ways I am looking at it right now. Certainly, the singular values of X^TX are ...
3
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1answer
46 views

If $\frac{1}{x^2+x+1}$, then find $y_n$ ($n^{th} $differention of the equation).

The answer: $$\frac{2(-1)^n \cdot n!}{\sqrt{3}r^{n+1}}\sin(n+1)\theta,$$ where $r=\sqrt{x^2+x+1}$ and $\theta=\cot^{-1}\frac{2x+1}{\sqrt{3}}$. How I have tried: ...
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0answers
30 views

Does linearity imply differentiability?

Is a linear function differentiable at every point by definition?
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0answers
21 views

On limit of function and differentiability on endpoints of an open interval

Before asking a question I would first like to mention the definitions of limit of function and differentiality at x=p 1) Limit of function (f) at x=p Let E be domain of f and p be a limit point of ...
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2answers
30 views

Show that a function is a solution to differential equation [duplicate]

I have a homogenous differential equation $a_0 y'' + a_1 y' + a_2 y = 0$ I know that $\lambda_0$ is a double root in characteristic polynomial. Now I have to show that $y(t) = t e^{\lambda_0 t}$ is ...
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0answers
10 views

Differential equation notation(syntax?) concern

My concern is not with the question itself but I guess if my "syntax" is correct. The question is A radioactive material, such as the isotope thorium-234,disintegrates at a rate proportional to the ...
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1answer
20 views

solution to an ODE separable variable

I have this ODE: $du/u = dt$ which I think the solution is $\ln|u| = t + c \leftrightarrow |u| = e^{t+c} = ce^t$. Is this correct? Is $u$ really a function then, since each value of $t$ would be ...
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0answers
23 views

Principle of solutions overlapping.

Please help me to prove that, if the functions y = f (x) and y = g (x) are solutions, respectively, of the non-homogeneous linear differential equations with constant coefficients: y" + a y' + b = ...
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2answers
25 views

Half-life time computation, and percentage of isotope remaining after X years.

The half-life of a certain radioactive substance is 1400 years. What is the percentage of radioactive isotopes still present after 700 years? The reference solution of my book is: 80.8% Thank you ...
1
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1answer
46 views

Find $ y''$ and $ y'''$ if $b^2x^2+a^2y^2=a^2b^2$

answers :- $$y''=-\frac{b^4}{a^2y^3}$$ and $$ y'''=-\frac{3b^6x}{a^4y^5}$$i have tried solve in this manner:-$$b^2x^2+a^2y^2=a^2b^2 -1.)$$diff wrt x :$$2b^2x+2a^2y'=0$$ ...
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0answers
26 views

Interval for a maximal solution

So, I have this multiple answers question from a test for analysis, and it's driving me crazy for either I don't understand correctly the concept of maximal solution, or the options given are wrong. ...
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0answers
21 views

using convolution for power series solution method for DE's

Say I have a homogeneous linear differential equation of the form $y''+py'+qy=0$ and I want to solve it using the power series solution method. So I use the substitution ...
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0answers
28 views

IF $P^2=a^2 .cos^2 \theta+b^2 .cos^2\theta$,prove that $ P+\frac{d^2 p}{d\theta^2}=\frac{a^2 .b^2}{P^3}$ [duplicate]

how i have done:- diff wrt $\theta$:- $2p\frac{d p}{d\theta}=a^2 (2cos \theta)(sin \theta)+b^2 (2sin \theta)(cos \theta)$ 1.) $2p\frac{d p}{d\theta}=b^2(sin 2\theta)+a^2 (sin 2\theta)$ double diff ...
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3answers
267 views

Is my solution to this differential equation correct?

My answer is: $[(1+x^2)^3]y = \dfrac{(1+x^2)^3}3+C$ But this option is not given, so is it correct? Thanks
2
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1answer
56 views

Finding the total differential.

My understanding of this is that it is a rate of change and in my mind I think of it like a vector.. you go some units in the $x$ direction and some units in the $y$ direction giving you the total ...
2
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1answer
53 views

How can I solve this differential equation, what type is it?

How can I solve this differential equation, what type is it? $$(x^2+2x-2y)dx=dy$$ How can I find the integrating factor?
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2answers
28 views

The general solution for this differential equation?

Find the general solution of this differential equation: $$ \frac{dy}{dx} = \frac{3x^5 y^3}{4} $$ Here's what I've done so far: $ dy=\frac{3x^5 y^3 dx}{4} $ $ 4dy = 3x^5 y^3 dx $ $ \frac{4dy}{y^3} ...
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1answer
21 views

Continuous extension of differential operator to sobolev spaces

If $T$ is an differential operator of order $k$ from $\mathbb{C}$-vector bundle $E$ to $F$ over a compact differential manifold $X$. Question: how can we extend it to a continuous linear map between ...
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3answers
146 views

Explanation of line element formula $dl^2 = dx^2 + dy^2$

I found this in a physics textbook without justification: $$dl^2 = dx^2 +dy^2,$$ where I presume that $l = \sqrt{x^2+y^2}$. Why is this so? By my calculations I obtain $$ dl = \dfrac{\partial ...
2
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1answer
42 views

Differential Form Over $S^2$

I was checking problems on differential forms and I found the following one. Consider the sphere $S^2 \subseteq R^3$ and the map $\omega_p : T_pS^2 \times T_pS^2 \rightarrow \mathbb{R}$ given by ...
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0answers
43 views

A New Take On The Snow Plow Problem

The problem: One day it started snowing at a heavy and steady rate. A snowplow started out at noon, going 2 miles the first hour and 1 mile the second hour. What time did it start snowing? I know ...
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1answer
37 views

Deriving FTC from the generalized Stokes.

How do I derive the Fundamental Theorem of Calculus from the generalized Stokes theorem?
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0answers
21 views

Two forms of application of metric tensor to get differential length

I'm reading the monograph "Foundations of Tensor Analysis for Students of Physics and Engineering With an Introduction to the Theory of Relativity" by Joseph Kolecki, now retired, of NASA. I have a ...
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1answer
63 views

Closed form solution to first order differential equation

I wonder why two solutions are not the same: The differential equation is: ds(t)/dt=A*s(t)+B(t) with initial condition s(0)=s0 A is a 2*2 constant matrix, s(t) is a 2*1 variable vector and B(t) ...
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1answer
31 views

Finding the constant for the particular solution to $y''(x) + y(x) = 2^x$

I'm really confused on this problem. Right now, I'm solving for the particular solution of: $y''(x) + y'(x) = 2^x$ My test solution was $A^x$, and I got $yp(x) = 2^x/(ln(A)^2 + 1)$ My problem here ...
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2answers
45 views

Finding points where a smooth map between differential manifolds is or is not an immersion.

I am having trouble answering questions pertaining to immersions on smooth manifolds. For example: Given the unit sphere $S^2$ around the origin in $R^3$ and the map $f: S^2 \rightarrow R^3$ given ...
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2answers
50 views

Finding the differential of $y=(u+1)/(u-1)$

I'm having trouble with differentials. I've been trying to learn about them online using great resources like PatrickJMT but I'm having trouble finding examples for this kind or problem. I hate asking ...
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1answer
27 views

Finsler Metric from page 2 of the book by Chern and Shen.

Physicist here not a mathematician. I am trying to understand the notation for the Finsler metric in Chern and Shen's book. The equation is $$\textbf{g}_y(u,v):=\frac{1}{2} ...
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0answers
32 views

n-form density weight of the levi civita tensor

Why does the n form of levi civita $\epsilon:$=$w_{1}\wedge w_{2} ...\wedge w_{n}$ have a density of weight -1 where $w_{i}$'s are cobasis of some vector basis? I thought this geometrical quantity ...
2
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2answers
42 views

Fresnel Integral multiplied with cosine term.

$$I=\int_a^b \sin(\alpha-\beta x^2)\cos(x)\, dx.$$ Can anybody tell me, how to solve this integral ? I know that this is related to Fresnel Integral if the $\cos(x)$ term is absent.
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3answers
110 views

Does $f(dx)$ have any meaning?

Simple question, does a differential $dx$ have any meaning composed in a function $f$, such as $\sqrt{dx}$, where $f(x)\neq x$?
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4answers
74 views

Differential of $x \sin x$

Finding the differential of $x \sin(x)$ \begin{align} d(x \sin x) &= (dx) sinx + x (d(\sin x)) &&\text{using the product rule}\\ &= \sin x\,dx + x \cos x\,dx \end{align} My question ...
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7answers
192 views

Is There a Difference Between $d^2x$ and $(dx)^2$?

I've just started reading through Calculus Made Easy by Silvanus Thompson and am trying to solidify the concept of differentials in my mind before progressing too far through the text. In Chapter 1 ...
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2answers
38 views

Solving a first order nonlinear ODE (nonseparable)

$$ f(x)=\int_{0}^{x}\frac{1}{1-af(t)}dt $$ How would one go about solving this equation? Does this equation have an analytical solution? I have only learned different methods for solving linear ODE ...
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0answers
13 views

Differential Formula Simplification

Define operators $x,D,1$ by $xf=xf$, $Df=\frac{d}{dx}f=f'$, and $1f=f$. Notice, then that $$(x+D)^nf=\sum_{k=0}^np_k(x)D^kf,\ \ \ \ \ \ \ f\in R[x],$$ for some sequence of polynomials ...
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1answer
29 views

About immersion of $ S^1$

I would like to ask if there is immersion $f: S^1 \to R^2$ that can not be $C^0$ approximated by embedding $S^1 \to R^2$. To be precise, If there exists $\varepsilon_0>0$, such that for any smooth ...
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0answers
39 views

Method to solve epidemic differential equation

I am searching for a method to solve this epidemic growth equation $$ \frac {dx(t)}{dt} = a (1 - u - x(t)) (x(t) + s e^{bt}) - q x(t) $$ with $a, u, s, b, q$ constants.
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1answer
27 views

Slope of a function very much less than the function

I was working on a Cosmology problem and got stuck at this approximation used in a paper. Fundamentally the approximation is, $\frac{df(x)}{dx} \ll f(x)$. Now I can't understand how to imagine this ...
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1answer
19 views

Finding the degree and order of differential equations

Find the order and degree of the differential equation $\mid \frac{dy}{dx} \mid + \mid y \mid = 0$
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0answers
37 views

Understanding notation with regards to tangent derivatives.

I am currently reading a paper on monge-ampere equations, and in one part the author does as follows. Let $\Omega,\Omega^*$ be two uniformly convex subsets of $\Bbb R^n$, and let $h\in C^{2,1}(\Bbb ...