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

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0answers
28 views

diagonal matrix and differential equations

I need help , be A= 1 0 0 1 0 2 0 0 0 0 1 0 1 0 0 1 real eigenvalues are : ${ 0 ; 1 ; 2 ; 2 }$ Eigenvectors: eigenvalue 0: $[ -1 ; 0;...
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1answer
25 views

Continuous and differential inverse function

I have a very interesting question: Given a function $f$ which is continuous but need not be differentiable. Then the correct statement is a. it can be an odd function b. it can't be an ...
-1
votes
0answers
35 views

Differential Equations confused over question

Interpret the statement as a differential equation. On the graph of y = \Phi(x) , the slope of the tangent line at a point P(x, y) is the square of the distance from P(x, y) to the origin. From my ...
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1answer
16 views

Condition for an expression to be a total differential

I have fully understood the concept and formulae around total differentials of multivariate functions. What is the condition however for an expression of differentials to be the total differential of ...
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0answers
34 views

differentials in physics [migrated]

Often I find the following expressions in physics books: Say we have a current density $\vec{j}=\rho\vec{v}$ through a surface $\vec{F}$ of particles $N$ in the volume $V$ with the density $\rho=dN/dV$...
4
votes
1answer
75 views

Leibniz rule; Solving differential equations

Could you help me with a question? I get stuck at ii), Define the function $$I(x):=\frac{1}{\pi} \int^\pi_0 \cos(x\sin\theta) d\theta$$ i) Via application of Leibniz rule (or otherwise) calculate ...
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0answers
27 views

Solving differential recurrence equations

I played around trying to make an equation describing Fibonacci numbers and ended up finding out that what I'd created was something called a recurrence equation: $f(x)=f(x-1)+f(x-2)$ ($f(x)$ is ...
1
vote
3answers
58 views

Linear 1st order differential equation

I tried to solve this equation but need help from seniors what to do next... $$\frac{dq}{dt} + q = 4\cos2t ; q(0) =1 $$ Multiplying both sides by I(t) i.e. $$I(t)= e^t$$ $$e^t\frac{dq}{dt} + e^t q =...
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0answers
49 views

Partial derivative of vector intercepting a plane

I was reading a paper that describes the partial derivatives of a range $\rho$ that intercepts an arbitrary surface, where $\rho = |\bar{r}_t - \bar{r}_{bf}|$. The author described the influence of an ...
0
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0answers
14 views

Is it possible and if so how do I solve this system of linear equations?

So I want to solve this system of 3 differential equations for S, I and R I've looked all over the internet but I guess I also have no idea how to search for a way to solve it. S'(t)=I(t)S(t)C I'(t)...
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0answers
21 views

Cauchy problem for a differential equation

I have the following Cauchy problem for the oscilator: enter image description here The texbook says: 1) If c=0.2 and a=0 solve symbolically (?) the problem and draw the graphic of the solution in ...
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0answers
13 views

Solving Linear System of Differential Equations and Checking Solution

$2x'-x+y'-y=e^{-t}$ $x'+2x+y'+y=e^t$ $D$ is the symbol used instead of writing $d/dt$. I used an elimination method to solve for $X_{general}$, in other words, From $(2D-1)[x]+(D-1)[y]=e^{-t}$ $(...
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1answer
12 views

Stationary points of a system of differential equations

I have the following system of differential equations: enter image description here The questions I have to answer are the following: 1) Find the stationary points of the system 2) Draw the vector ...
2
votes
2answers
44 views

What is the logic behind decomposing a derivative operator symbol. In the population growth equation? [duplicate]

Why is this algebra calculus trick legal? $$ \frac{dy}{dt} = Ky$$ $$ \frac{1}{y}\frac{dy}{dt} = K$$ ...then my teacher did something illogical he decomposed the Differential operator $\frac{dy}{...
2
votes
1answer
18 views

Jacobian determinant of a map?

For $m,n\in \mathbb N$, let $f$ is the map given by $$\begin{align} f: & \quad \mathbb R^m \times \mathbb R^n \longrightarrow \mathbb R^m \times \mathbb R^n \\ & (x,y)\mapsto f(x,y) = (x+x',...
0
votes
0answers
60 views

Principal Symbol

I'm trying to prove the claims about the symbol of a differential operator. What I did was use the Leibniz formula to calculate the symbol of the composition. If $P u(x) := \sum_{|\alpha|\leq m} a_\...
0
votes
0answers
30 views

Multiscale asymptotic expansion of differential equation with a constant

My question is: How to apply a two-time asymptotic analysis to the equation: $\frac{\partial v}{\partial t} + v\frac{\partial v}{\partial x} = \beta+\,\,\frac{d I}{d x}$ $\,\,\,\,\,\,\,\,\,\,...
0
votes
2answers
33 views

Differential equation, determinant = 0

$$(x+2y+1)y' = 2x+4y+3$$ So i've tried to write it like: $$y' = \frac{(2x+4y+3)}{x+2y+1}$$ and when i do the determinant i get that its $0$. I've tried to write it then with $\alpha$ and $\beta$ ...
-1
votes
0answers
25 views

Differentiable function f(x)

Let $f(x)$ is a differentiable function satisfying $f'(x) + 100 f(x) ≤ 1 $ Then $f(x) -1/k$ is a non increasing function of $x$ , then we have to find the value of $k $ I tried , but at last ...
0
votes
4answers
39 views

Finding the polynomial

Find a polynomial function $p(x)$ such that $p(2x)=p'(x)p''(x)\not=0$
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0answers
37 views

Clarification of meaning of dx in an integral [duplicate]

I would like to have some clarification on the physical meaning of $dx$. I already know the following in the context of the area under the curve: $\lim_{\Delta x \rightarrow 0} \sum f(x) \Delta x \...
0
votes
0answers
18 views

How do I determine value of a single attribute from a cluster of attributes?

Given price and demand for a cluster of attributes, how can I solve for a single attribute differential? For example, Hotel rooms with Standard Layout and King Bed have an average rate of $150 and ...
0
votes
0answers
29 views

How to apply the chain rule to a=x-ct and b=x+ct to get this result?

Given $$a=x-ct \quad and \quad b=x+ct$$ My textbook says, using the chain rule, we can derive: $$\frac{\partial}{\partial x}=\frac{\partial}{\partial a}+\frac{\partial}{\partial b} \quad and \quad \...
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votes
2answers
40 views

Surface integral of a scalar over a unit cube.

Evaluate the following integral $$\iint_S (x+y+z) \, dS$$ where $S$ is the surface of the cube $[0,1] \times [0,1] \times [0,1]$ Honestly, I don't know what to do. All I know is that you have to ...
0
votes
1answer
17 views

Second order differential equation particular solution of a product

I read that when the right side $f(x) = e^x$, a suggested form of $y_{ps}$ is $Ce^x$, and for $f(x)$ is linear in $x$, a form of $y_{ps}$ is $Cx + D$. If $f(x)$ is the product of the two mentioned, ...
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votes
5answers
50 views

Show that f(x) = $\sin(\frac1x)$ is not differentiable at $x=0$

I've been looking forever and have yet to find any examples of someone actually working out the limit of this problem: $$\lim_{x\to0} \sin(\frac1x)$$ I'm stuck at the beginning: $$\lim_{h\to0} \...
0
votes
1answer
44 views

calculating coordinates along a clothoid betwen 2 curves

I need to write a program that will calculate coordinates along the designed rails of a proposed subway. The designed polyline representing the rail is composed of lines, arcs (circular curves) and ...
0
votes
0answers
39 views

How to find the solutions for these constraints?

$r(x)$ and $p(x)$ are two functions of $x$, and $0.5\le x\le 1$. They have the following properties: \begin{equation} \begin{split} &\frac{r'(x)}{p'(x)}=\frac{1}{x}-1\\ &r''(x)\cdot ...
2
votes
0answers
26 views

Under which conditions is $f(x)=\frac{1}{2}x^TPx+q^Tx+r$ convex?

I am given the function $$f(x)=\frac{1}{2}x^TPx+q^Tx+r$$ and am asked to establish under which conditions $f(x)$ is a convex function. I have to use the definition of a convex function where we look ...
1
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1answer
16 views

Alternative expression for the differential solid angle?

Attached clipping from my lecture notes. In this expression for the differential solid angle element I don't quite see how: $$ \sin\theta \, d\theta=d(\cos\theta) $$ Why is it not: $$ -\sin(\theta) \,...
1
vote
2answers
36 views

Solve first 1st order differential dV/dt=S-CV^1/2

I am trying to solve the following first order differential equation: $\frac{dV}{dt} = S-CV^\frac{1}{2}$ I can't see how this could be solved via separation of variables or integrating factor as it ...
1
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1answer
19 views

Factorization of a smooth function problem

Let $f:\mathbb{R}^n \rightarrow \mathbb{R}$ be a smooth function such that $f(0) = 0$ and $\nabla f(0) = O$. It is possible to factorize $f$ as $f(x) = g(x) h(x)$ where $g,h :\mathbb{R}^n \...
0
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0answers
39 views

Non differentiable function

Let $\varphi:\mathbb{R}\rightarrow \mathbb{R}$ and $f:\mathbb{R}\rightarrow \mathbb{R}$, be functions defined by $$\varphi (x)=\vert\sin(\pi x)\vert \ \ \ x\in\mathbb{R} $$ $$f(x)=\sum_{i=0}^{i=\...
0
votes
4answers
61 views

Dividing derivatives by derivatives

We are often taught that $$\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{dy}{dx}$$ Why are we allowed to say this? What about the case of higher derivaitves, i.e. $$\frac{\frac{d^ny}{dt^n}}{\frac{d^nx}{...
0
votes
1answer
26 views

Find the general solution to the ode?

$y′′+5y′+6y=85 e^{6x}\cos2x$ What is the general solution to this? I got.. $y(x)=Ae^{−2x}+Be^{−3x}$ then doing it again i got a different answer. $y(x)=Ae^{−x}+Be^{−2x}+86e^{2x}\sin3x$ What is ...
0
votes
1answer
22 views

Find a solution to the ODE?

$dy/dx = x\cos2x/3y^2$ So far I've rearranged. $dy3y^2 = x\cos2x dx$ Then do I just solve for $y$? If so how do I do that? I'm just a little confused on the next steps. Thanks for any help.
0
votes
0answers
27 views

Particular solution to system of linear differential equations for singular system matrix

I have a linear system on standard state space form $\dot{x}(t) = Ax(t) + Bu(t)$ I would like to find the solution to this, which usually is $x(t) = e^{At}x(0) + \int\limits_0^t e^{A(t-\tau)}Bu(\...
1
vote
2answers
43 views

Differential Equation - Where does the solution end?

I was asked to solve the differential equation $y'+\frac{y}{x+1}=\frac{2y-1}{x}$, given the starting point y(0.5)=5/6. The equation meets the criteria for Existence and Uniqueness for every x>0 (as y' ...
0
votes
0answers
17 views

Minimal and maximal problem

Cuboid with contlstant volume $V$ its base is a rectangle length $nX$, width $X$ The whole area of cuboid is $A$ and $A$ was minimal area Prove with differentiation that $A^3n=54 V^2(1+n)^2$
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1answer
31 views

Differential geometry, equation, exercise problem

There is a problem in the text " O'neill- Elementary differential geometry 2ed. In my opinion, some expressions in (a) are not correct. What's the exact meaning of following statement? " the ...
0
votes
1answer
29 views

Differential equation with constant coefficients

When we solve differential equation with constant coeffiecients we find out the auxiliary equation, then its roots and proceed further. But my question is When for example we have an equation of the ...
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votes
1answer
33 views

Differentiation of a trig function [closed]

Please help me to differentiate following equation $$ y=\frac{\cos3x}{\sin2x} $$
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votes
0answers
17 views

Integration factor of a function of X

$(-3x^2-5y)$ $dx$ + ($-3x)$ $dy$ $=0$ $M_y$$=-5$ , $N_x=-3$ The equation is not exact, so I now will find an integrating factor using the formula $$(\frac{M_y-N_x}{N})$$ $M_y-N_x$= $-5-(-3)=-2$, $\...
0
votes
1answer
39 views

Find the general solution to the differential equation $y″ + 4 y′ + 5y = 0$

I'm not sure if I am doing this right but I have the characteristic polynomial as $r^2 + 4r + 5$, which factors to give the roots $-2 +- i$. I then get $y_1(x) = c_1 e^{(-2+i)x}, y_2(x) = c_2 e^{(-2 -...
1
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1answer
53 views

Find the general solution of $y' \cot x +y =2$

How do you start by finding the general solution? And then finding the integration constant using the initial condition $y(0)=1$ So far I've got... $$y' \cot x +y = 2\\\frac{dy}{2}-y = \tan x dx\\\...
1
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0answers
31 views

Differential dimension of $W^{l,p}(\mathbb{R^n})$

I'm looking for the differential dimension of $W^{l,p}(\mathbb{R^n})$, defined as follows: Given a function space $Z(\mathbb{R^n})$, we say that $\mu\in \mathbb{R}$ is the differential dimension of $...
0
votes
1answer
40 views

Find the differential of this function?

$$y'= \left(\frac{y+2}{x+y-1}\right)^2$$ So, i've been struggling with this for about 4 days, and sure, call me names, math is not my stronger suit, but I'd really like to figure this out. I've tried ...
0
votes
3answers
28 views

How do I find the laplace transform of a product? [closed]

How do I find the laplace transform of a product? Specifically $e^{5t}\cos{t}$?
6
votes
4answers
651 views

Continuity of Derivative at a point.

Is it possible that derivative of a function exists at a point but derivative does not exist in neighbourhood of that point. If this happens then how is it possible. I feel that if derivative exists ...