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

learn more… | top users | synonyms

0
votes
0answers
16 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 $A^3n=54 V^2(1+n)^2$
0
votes
0answers
17 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
0answers
4 views

derivation of 3rd order Adams method

can anyone explain me the derivation of 3rd order Admas method, like how they derive the equation if this answer is done then second part is that why we have 1 to 4 order Adams formula..are they more ...
-3
votes
0answers
35 views

Differential calculus: function of two variables question (four parts) [on hold]

Let the function of two variables be given as $f(x, y) = \sin(x + y), x ∈ \mathbb{R}, y ∈ \mathbb{R} $ (a) Sketch the surface $z = \sin(x + y)$ and draw the level sets of this function for levels ...
-2
votes
0answers
27 views

Easy Differential Equation [closed]

I want to solve this easy differential equation : $y'(t)=y(t+b)$ with b a real number and y is a $2\pi$ periodic function. In my exercise , we use Fourier coefficients to solve it but there is not ...
0
votes
1answer
26 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 ...
-1
votes
1answer
30 views

Differentiation of a trig function [closed]

Please help me to differentiate following equation $$ y=\frac{\cos3x}{\sin2x} $$
0
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
37 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
vote
1answer
50 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 ...
1
vote
0answers
29 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
35 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
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
629 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
votes
2answers
56 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
votes
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
votes
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 ...
0
votes
0answers
4 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
27 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
votes
0answers
26 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 ...
1
vote
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
vote
1answer
26 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
vote
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 $X'(0)$.
1
vote
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
vote
0answers
25 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
236 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. $y' = \frac{dy}{dx}$ ...
-1
votes
0answers
53 views

Help to find solution of differential equation

I'm trying to solve this differential equation, but cannot, please help me to find answer. $Z' + p(x) Z^\alpha = g(x) Z^{ \alpha - 1}$, where $Z'$ : the first derivative and $\alpha \in (0, 1)$.
0
votes
0answers
48 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? ...
1
vote
1answer
34 views

Differential of Fisher-Weil Bond Price using discrete-time compunding

I would like to ask you if my calculations are correct regarding the differential of the price of a bond using a sequence of spot rates as discount factors. \begin{equation} P(i^S ) = \sum_{t=1}^{T} ...
0
votes
0answers
17 views

Differentiability of multivariable functions.

I have the following two questions: (1) What are the most important techniques to show if a multivariable function is differentiable. (2) I know how to show that a multivariable function is not ...
0
votes
1answer
14 views

Expressing Δf with an integral of a differential (infinitesimal)

My Thermodynamics teacher used the relation in the image and I would like to know how can I prove it! http://i.stack.imgur.com/EQlB4.jpg Thank you!
1
vote
2answers
33 views

Find the solution of this integral

I have to solve this differential equation: After following the necessary steps, I have come to a point where I can't solve this integral: Could you please help me?
0
votes
1answer
24 views

Finding Constants on a Differential Equation

Question (from my sample exercises in calculus): Find constants $A, B$ and $C$ such that the function $$ y=A\sin x+B\cos x $$ satisfies the differential equation $$ y''+y'-2y=\sin x. $$ I am ...
2
votes
1answer
54 views

Spivak “Differential Geometry ”volume 1 page 41

I want to ask question related to Lemma 6 from Spivak "Differential Geometry Volume 1" page 41. The lemma states: If $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is $C^{1}$ and $A\subset\mathbb{R}^{n}$ ...
1
vote
0answers
43 views

Differential Equation $xf''(x) + 3x(f'(x))^2 = 1 - e^{-x}$

Given a function $f$ which satisfies the differential equation $xf''(x) + 3x(f'(x))^2 = 1 - e^{-x}$ if $f(0) = f'(0) = 0$, find the smallest constant $A$ such that $f(x) \le Ax^2$ for all $x \ge 0$
0
votes
1answer
23 views

Find the Eigenvalues of $xy''+y'+λy=0, y(1)=5, y(e)=2$

Hi everybody I have to Find the Eigenvalues of $xy''+y'+λy=0,$ in $y(1)=5, y(e)=2$ I think it has to be in the Stourm-Liouville form: $d/dx(xy')+λy/x=0$ but Im not sure about this
0
votes
1answer
30 views

Joint and Marginal Probability Densities

I have essentially two questions. How should we treat a probabilistic differential given an intergral? For example, \begin{align} \text{Compute } \int_0^a u &dF(u|x), \text{where} \\ U &\sim ...
0
votes
0answers
18 views

Accounting for the superposition of waves via a differential equation

$L_y=f_1(t)+f_2(t)+f_3(t)$ If $L_y=f_1$ is solved by $y_1$, $L_y=f_1$ by $y_2$, and $L_y=f_3$ by $y_3$, then I've reasoned that a good general solution for such a differential equation would be ...
1
vote
0answers
18 views

Distribution functions: differentials in the numerator or denominator

One paper I'm looking at says, $n(M, z) \, dM \, dz$, the number of sources with mass $M$ at a redshift $z$, in the mass interval $dM$ occurring in the redshift interval $dz$. While another ...
0
votes
1answer
37 views

Differentiable in real analysis problem

Let $f$ be a real value function that is differentiable in $[a,b]$, with . Show that there is exists $x_0$ in $(a,b)$ such that: I don't know how to prove it. Please, could anyone give me a hint. ...
1
vote
2answers
50 views

Foundations of Differential Calculus

In the preface of Foundations of Differential Calculus there's a section that says: Thus, if the quantity $x$ is given an increment $\omega$, so that it becomes $x + \omega$, its square $x^2$ ...
3
votes
2answers
62 views

Matrix derivation definition

I am tring to compute a mathematical derivation, but I am obviously missing something. I precise that I have only learned "formal" definition of derivation in the 1D case, and am not familiar with ...
0
votes
1answer
14 views

Matrix exponential map differentiation.

Given any matrix $A$, one wishes to give an expansion at $A$, and especially wishes to unravel a possible differentiation of the $\\\exp: Mat(n, \mathbb{C}) \rightarrow GL(n, \mathbb{C})$ mapping. ...
-2
votes
1answer
60 views

Tumor Growth Model. Differential Equation [closed]

I have a calculus question that I can't figure out. Thank you in advance for the help! A model for tumor growth is the Gompertz function that is a solution to the differential equation ...
-1
votes
2answers
55 views

differentiate by a differential

This is a difficult question to phrase so I will show it mathematically. I am trying to differentiate something through chain rule but I am not sure if my steps are correct. Let $f(\theta) = ...