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

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4answers
35 views

Finding the polynomial

Find a polynomial function $p(x)$ such that $p(2x)=p'(x)p''(x)\not=0$
1
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0answers
34 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 ...
-1
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0answers
13 views

Jacobian Matrix Surface Patches [on hold]

How can I find Jacobia matrix of F=$inv(\tilde{\sigma})$ 0 f o ${\sigma}$, for (f o ${\sigma}$ ) ($\theta$,$\phi$) = ...
0
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0answers
17 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
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0answers
28 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 ...
0
votes
3answers
36 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
13 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, ...
0
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0answers
10 views

Prove that P(x,y)+Q(x,y)=0, will have a integrating factor of form phi(x+y), if 1/(P-Q)*(del delx of P- del delx of Q)is of form x-y. [closed]

Prove that P(x,y)+Q(x,y)=0, will have a integrating factor of form phi(x+y), if 1/(P-Q)*(del delx of P- del delx of Q)is of form x+y.
-1
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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
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1answer
27 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
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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
15 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
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2answers
34 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
18 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 $$\varphi (x)=\vert\sin(\pi x)\vert \ \ \ x\in\mathbb{R} $$ ...
0
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4answers
58 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. ...
0
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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.
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0answers
26 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 ...
1
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2answers
41 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
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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
30 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 ...
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1answer
28 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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1answer
33 views

Differentiation of a trig function [closed]

Please help me to differentiate following equation $$ y=\frac{\cos3x}{\sin2x} $$
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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
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1answer
38 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
52 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 ...
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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
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1answer
37 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
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3answers
24 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
647 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 ...
0
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2answers
57 views

Differentiating the term, $y=\sqrt{\frac{1-x^2}{1-x}}$

In my calculus book, the above question was given. Since the the term can be simplified, I simplified it to $$\sqrt{(1+x)}$$ which differentiates to $$0.5\sqrt{{\frac{1}{1+x}}}$$ But the answer in the ...
0
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0answers
21 views

Is this solution of the PDE correct?

http://www.impa.br/opencms/pt/ensino/downloads/mestrado_profissional_projeto_fim_curso/projetos_fim_cursos_2010/Diogo_Duarte.pdf Pages 24-25 How do they get from 2.30 to 2.32 using boundary ...
0
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1answer
19 views

function representation of power series

What is the function representation of this power series? [Summation from n=0 to infinity of ($x^n)(n+1)!/n!$ The solution is $\frac{1}{(1-x)^-2}$ but how??? I know that ...
0
votes
2answers
29 views

Try to approximate this function at these three points for a deviation Δx =0.1

Take the function y=x^2 . Take three points, say x=0, x=1, x=3. Try to approximate this function at these three points for a deviation Δx =0.1. For which of the three points the approximation works ...
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0answers
5 views

Determine $(g \circ f)'(1, 1)$ and an approximate value of $(g \circ f)(1,01; 1,01)$. Approximation by differentials and chain rule.

First time posting. Excuse me for the formatting or grammar. Question Let $f: R^2 \to R^3$ be a differential function, such that $f(1,1) = (3, 1, 2)$ and $f'(1,1): R^2 \to R^3$ is given by the ...
3
votes
2answers
90 views

What is integral of a function of a differential?

What is $$\int \dfrac{d\theta}{\sin\frac{d\theta}{2}}$$ I thought of approximating the $\sin$ term $$\sin \dfrac{d\theta}{2} \approx\dfrac{d\theta}{2}$$ and so the integral evaluates to be $$\int ...
0
votes
1answer
49 views

Question about differentiable function

Let $f:[0,1]\to R$ be a real-valued function which is continuous on $[0, 1]$, differentiable on $(0,1)$ and $f(0)=f(1)=0$, $f(1/2)=1$. prove the following statements $a$. There exists $\epsilon \in ...
0
votes
0answers
28 views

Solving $2ty''+y'+2ty=0$ centered at 0 using the method of Frobenius.

This is what I've got so far: $$2ty''+y'+2ty=0$$ $$y''+\frac{1}{2t} y + y = 0$$ $$\sum_{n=0}^{\infty}(n+r-1)(n+r)c_n t^{n+r-2} + \frac{1}{2t}\sum_{n=0}^{\infty} (n+r)c_n t^{n+r-1} - ...
0
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0answers
27 views

Using Fourier Series to find formulas for f(x)

Given $$ f(x) = \begin{cases}x+1,&-1<x<0\\x,& 0<x<1\end{cases} $$ and $$ f(x+2)=f(x), $$ I am asked to find the formula for $f(x)$ in the intervals $1<x<2$ and ...
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2answers
40 views

Where am I fault? Find g(x) when $\lim_{h\rightarrow 0}\frac{g(x+h)-2g(x)+g(x-h)}{h^2}=20x^3+6x.$

I got some problems here and I can't understand where am I fault. The task says: if $g$ is 2 times differentiable $,g'(0)=g(0)=1,$$$g''(x)=\lim_{h\rightarrow 0}\frac{g'(x)-g'(x-h)}{h},$$ and also ...
1
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1answer
27 views

Having trouble with an Eigenvalue Differential Equation

Here is the problem: $$ x^2y''-xy+\lambda y = 0,\quad y(1)=0,\quad y(L)=0,\quad L>0 $$ I am asked to find the Eigenvalues and Eigenfunction. I can't figure out how to get a general equation for ...
1
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0answers
30 views

System of differential equations of second order $X''(t) = -(A^2)X(t)$

How do I solve problems of the form $$X''(t) = -(A^2)X(t),$$ where $X$ is a $2 \times 1$ matrix and $A$ is a $2 \times 2$ matrix? We're given $X(0)$ and $X'(0)$.
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0answers
34 views

Help solving this 2nd order non-liner differential equation?

I am trying to compute the optimal path length for getting from Sydney to Hong Kong via a tunnel and using only the force of gravity. (See figure.) By using calculus of variations and the ...
1
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0answers
26 views

Derivative multiple variables and summation

I ran into a pretty nasty differential equation where we have the following: Differential in respects to k and I'm looking to seperate K here. so what I recall from doing these 'in respects to' ...
0
votes
1answer
24 views

Differential Equation and subspace of solution

Show that the solutions of the homogeneous linear ODE $$\frac{dy}{dx} + p(x)y = 0$$ on an interval I = [a,b] forms a vector subspace W of the real vector space of continuous function on I. What is the ...
1
vote
2answers
237 views

Ordinary Differential Equatiom

$$\left(1-x^2\right) y''-4 x y'-\left(1+x^2\right) y=x $$ I am required to solve the above differential equation. Can't get around how to approach. Any help would be appreciated. $y' = \frac{dy}{dx}$ ...
0
votes
0answers
57 views

Forming Differential Equations

Find the DE of the family of circles in $xy$ plane passing through the points $(−1,1)$ and $(1,1)$. The question was asked earlier also Finding the DE of family of curves but the answer was a tad ...
1
vote
0answers
27 views

Derivatives with rational orders like y^(1,5)'

Is there any mathematical definiton of rational derivatives? We all know y,y'',y''',...,y^(n) are orders of discrete numbers. Is there anything like one and a half 1,5th derivative of a function? ...