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

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4
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1answer
34 views

Definition second differential of a vector field

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a smooth function. Then we know that its differential $df: \mathbb{R}^2 \rightarrow Hom(\mathbb{R}^2,\mathbb{R}^2)$ maps vectors to matrices/linear ...
0
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2answers
23 views

Differential $\mathrm{d}^2f$ of implicit function $F(x,y,z)=xyz-x-y-z=0$

Determine the differential $\mathrm{d}^2f$ of the implicit function defined as $z=f(x,y)$: $$F(x,y,z)=xyz-x-y-z=0$$ So in fact of the implicit function I have to use the implicit function ...
0
votes
3answers
79 views

A paradox in differential calculus

Say I have a function $f=f(x,y)$ where $x,y$ are independent variables. Now, it is given that $p=x+y$. It can be shown that, since $x,y$ are independent, we get $$\frac{\partial p}{\partial x}=\frac{...
0
votes
1answer
36 views

Laplace Transform

Question: Use Laplace Transform to solve the following differential equation $\ y''+y =sin(t); y(0)=1, y'(0)=-1 $ My try,where F(s) is the transform of f(t)=y(t) $F(s)= \frac{1}{(s^2+1)^2} + \frac{...
1
vote
3answers
32 views

Inhomogenous Differential system

This is the system, $$ y'_1 =y_1+y_2+1 $$ $$ y'_2= -y_1+y_2+1 $$ initial value problem which fulfill: $$y_1(0)=1$$ $$y_2(0)=-1$$ value to find $$y_1(π)= \text{?}$$
1
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2answers
22 views

Differential system with initial value problem 2nd order

i got a problem solving this Diff. system with initial value problem 2nd order. $$ y''_1=−10y_1+6y_2 $$ $$y''_2=6y_1−10y_2$$ $$y_1(0)=1,y_2(0)=0,y_1'(0)=0,y_2'(0)=0 $$ i need the value for: $$ y_2(...
0
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0answers
16 views

Fiber contraction theorem in Ordinary Differential Equations

Let $\Sigma$, $\Gamma$ be metric spaces, with $\Sigma$ a complete metric space. Let $\phi \colon \Sigma \times \Gamma \to \Sigma$ such that $\exists \lambda \in ]0,1[$ and: $$ d(\phi (x_1,\tau), \phi(...
2
votes
0answers
31 views

solving differential equation [closed]

solve: $(1+p)^2r-2(1+p+q+pq)s+(1+p^2)t=0$ how do I solve this differential using monge's method? .Can someone help me with this
1
vote
1answer
37 views

How to derive this equation in many ways?

Given the equation: $$y=\sin^4(x)\cdot\cos^4(x)$$ I have derived the equation using product rule: $$f'(x)\cdot g(x)+f(x)\cdot g'(x)$$ Then I`ve got $$y'=4\sin^3(x)\cos^5(x)-4\sin^5(x)\cos^3(x)$$, then ...
0
votes
5answers
66 views

Find the derivative of $y=\sin^2(3x)$

Our instructor had given us an equation and we should get the derivative of it. But even him, he is confused what to do (funny college instructor) if we should use identities before deriving or either ...
3
votes
2answers
52 views

Sum of square of function

If $f'(x) = g(x)$ and $g'(x) = - f(x)$ for all real $x$ and $f(5) =2 =f'(5)$ then we have to find $f^2$$(10) + g^2(10)$ I tried but got stuck
0
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0answers
20 views

How to take derivative of boundary conditions of non linear ODE

I have a nonlinear boundary value problem to solve. Boundary conditions are given as r(x(t0),x(tf))=0 and I need to take the derivative of r with respect to x(to) and x(tf) respectively. I am ...
-1
votes
1answer
35 views

Can we convert first order ode to second order ode?

Is there any possibility to convert first order differential equation to second order differential equation? I have a system of first order differential equations as below and i need the right hand ...
2
votes
3answers
105 views

What is $dx$ on an integral? [duplicate]

I've heard from some of my teachers it's a bilineal form and some other stuff, nobody actually ever explained me the reason of it. Of course i've done practical problems in which $dx$ is a "very small ...
2
votes
1answer
40 views

What are the grounds for treating 'dx(differential, infinitesimal)' as if they were numbers?

I'm studying calculus and sometimes I find it strange to treat dx(differential) like numbers! Substitution rule would be a good example. ( I will use the first example in this website http://tutorial....
1
vote
1answer
19 views

Differential and lone $dx$es and the dependence on a free parameter

Suppose I get the an equation like $$ \frac{ds(t)}{dt} = \sum_i \frac{ds_i(t)}{dt} ,$$ as the result of some computation. In order to simplify this further I invoke the notion of the differential ...
2
votes
2answers
29 views

Baricenter of a region bounded by a closed parametric curve

I've always known how to get the center of mass of any region, but now i met a new question with a region bounded by a parametric curve and the question is to get its baricenter! My question is what ...
19
votes
10answers
2k views

What does it mean when dx is put on the start in an integral? [duplicate]

I have seen something like this before: $\int \frac{dx}{(e+1)^2}$. This is apparently another way to write $\int \frac{1}{(e+1)^2}dx$. However, considering this statement: $\int\frac{du}{(u-1)u^2} = \...
-2
votes
0answers
31 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;...
0
votes
1answer
26 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
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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 ...
4
votes
1answer
76 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 ...
1
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0answers
28 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 =...
0
votes
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
votes
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)...
0
votes
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 $(2D-1)[x]+(D-1)[y]=e^{-t}$ $(...
0
votes
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
19 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
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0answers
31 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 $\beta$ ...
0
votes
4answers
39 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
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
32 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
2answers
42 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
18 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, ...
-1
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
70 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
vote
1answer
17 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
vote
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.