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

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0
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0answers
21 views

Show that the normals to a parameterized curve all pass through the z-axis

I've been asked to show that the normals to a parameterized surface given by: $x(u,v) = (f(u)cosv,f(u)sinv,g(u)), f(u) \neq 0, g'(u) \neq 0$ all pass through the z-axis. I've computed the normal to ...
0
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1answer
34 views

differential equations solvable only by numerical methods [on hold]

What kind (a general formula would be nice) of differential equations do not have solutions expressible explicitly or implicitly or by an integral sign? In other words, what kind of differential ...
1
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2answers
27 views

Nonseparable differential equations

How can I solve the equation $\frac{dy}{dx} =\frac{x^2-y}{x-y^2}$? I've tried few substitutions such as $y=xv$ and $y=x/v$ but all to no avail! Please, help.
0
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1answer
15 views

Constant solutions and uniquenss of solutions theorem for IVPs

What role do constant solutions play in the existance and uniqueness theorem? For instance, consider the IVP $$\frac{dy}{dx} = x$$ $$ y(0) = 0 $$ Clearly, this IVP has a solution in the form of $y ...
0
votes
1answer
39 views

Equivalence of Gradient Fields and Exact Differentials on a Non-Simply Connected Region

I've recently been taking 18.02sc Multivariable Calculus on MIT OpenCourseWare, which states the following in one of their course notes: $$M \hat i + N \hat j = \nabla f \implies M dx + N dy \text{ ...
1
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0answers
52 views

Why is $\frac{\partial }{\partial y}\int M dx = \int \frac{\partial M}{\partial y}dx$

$M$ is a function of $x$ and $y$. I'm getting this question from looking at the solution of the exact equation $M \mathrm{dx} + N\mathrm{dy} = 0$.
0
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0answers
36 views

Independence of the choice of base for the differential.

Let $f:\mathbb{R}^m\longrightarrow \mathbb{R}^n$ be a mapping, defined by differentiable functions which, generally speaking, are non-linear and map zero into zero: ...
2
votes
1answer
32 views

Solving this 2nd Order non-homogeneous PDE

I am trying to solve the following equation: $$3u_{xx} - 10u_{xt} - 3u_{tt} = \sin(x + t)$$ I know that the left hand side is a quadratic equation which I have to factorise. Then I let one of the ...
0
votes
0answers
35 views

Does anyone use the notation $\mathrm d^3(x,y,z)$?

I am working on a LaTeX package for typesetting differentials (yes, I know it will be only one of many). I aim at covering many different situations that may arise when dealing with differentials. I ...
2
votes
3answers
51 views

differentials that I can't solve correctly

I can't solve these differential, someone can help me with a step by step solution? thanks $$y'+ty=t^3$$ $$y'=3t^2y+4t^2$$ I tried the first integrating by $$e^{\int tdt}$$ using $$p(t)=t$$ and ...
0
votes
2answers
32 views

Differential Equation Solution Incorrect

$$ty'= 3t^2-y$$ I can't solve this equation, someone that can help me with a solution step by step? my result is $\dfrac ct+t^3$ but the correct solution is $\dfrac ct+t^2$
1
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0answers
21 views

Derivative of a determinant with respect to a matrix

Can someone tell me the derivative of the following determinant ($\Psi\in\mathbb{R}^{p\times p}$, $Z\in\mathbb{R}^{p\times q}$, $\alpha\in\mathbb{R}^q$) $\frac{\partial}{\partial \Psi} ...
-1
votes
3answers
40 views

Differential equations: error on founding solution

I have to found the solution of this differential equation: y'=-2-y^2, it look simple but I didn't think so. After solving the differential eqaution I have to plot the solution. What I do: function ...
-1
votes
2answers
26 views

Differential Equations applications in Computer Science

I'm writing a project on differential equations and their applications on several scientific fields (such as electrical circuits, polulation dynamics, oscillations etc) but i'm mainly interested in DE ...
0
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1answer
17 views

Ordinary Differential Equations by Morris Tenenbaum and Harry Pollary

On definition 2.68, the book states that a set in the plane is called a region if it meets two conditions (p. 14): "Each point of the set is the center of a circle whose entire interior consists of ...
-3
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0answers
36 views

Find the integrating factor and solve for the equation.

$\left(2xy^2-y\right)dx + \left(2x-x^2y\right)dy= 0$ $2\,dx + \left(2x-3y-3\right) dy = 0;\quad y\left(2\right)= 0$ $\left(2y\sin\left(x\right)+3y^4\sin\left(x\right)\cos\left(x\right)\right)dx - ...
0
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2answers
31 views

How many tangents can be solved in this problem and how can I find it? [closed]

How many tangents can be solved in this problem and how can I find it? $y=x^3-7x+6$ at its points of intersection with the x-axis.
-2
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1answer
31 views

What is the equation of the tangent and normal at this problem? [closed]

What is the equation of the tangent and normal at this problem? Problem: $y=x^2-2x$ at its points of intersection with the line $y=3$. Use differentiation.
1
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1answer
14 views

How do I show that f(x) is independent for the variationproblem?

The variationproblem where f(x) is a reel differential function defined in the interval [0,1].
-3
votes
1answer
48 views

Differential Forms: Show that $d^2 = 0$ by explicit computation [closed]

Show that $d^2 = 0$ by explicitly computing $d^2\omega$ for $\omega$ a $1$-form in $\mathbb{R}^3$.
0
votes
1answer
38 views

How to find the solution to the differential equation $dy/dx$

How do I find the solution to the differential equation: $ \frac{dy}{dx} = e^{α(x+y)} + 3e^{αy}$? where $\alpha$ is not zero.
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0answers
11 views

2 dimension riemann manifolds of signature 0 metric

Does anyone have a proof that any 2d riemann manifold is conformally flat if metric has signature 0? Thanks.
-1
votes
0answers
38 views

Differential and partial derivatives : is it OK to divide by $ \mathrm dT $?

I have arrived at the equation : $$n (C_p - C_v ) \mathrm d T = T \left( \frac{\partial P}{\partial T} \right)_V \mathrm d V + T \left( \frac{\partial V}{\partial T} \right)_P \mathrm d P$$ I am ...
1
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2answers
40 views

What makes a differential equation, linear or non-linear?

Among these differential equations why one is linear while other is non-linear? What is criteria to find out whether a differential equation is linear or non-linear?
0
votes
2answers
31 views

Find the rate of change of the frequency when D, L, σ and T are varied singly.

I'm reading Calculus made easy to learn the notation (I know derivatives with the limit/prime style) and also some integral calculus which I haven't seen at school yet. You can check it here: ...
0
votes
1answer
50 views

Jacobian for matrix function involving kronecker product

I would like to ask you a little help for the following problem. Let $\Phi$ and $\Sigma$ be two $N \times N$ matrices s.t. the inverse of $(I_{N^2}-\Phi \otimes \Phi )$ exists and $\Sigma$ is ...
0
votes
1answer
31 views

PDE model of metal rod at temperature=1 plunged into a bath of temperature=0

Consider a metal rod (0 < x < l), insulated along its sides but not at its ends, which is initially at temperature=1. Suddenly both ends are plunged into a bath of temperature=0. Write the PDE, ...
1
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0answers
24 views

Given a set of arbitrary data, is it possible to model this data using differential functions.

Problem At the moment, I have a problem with seven variables: $S, A_1, A_2, R_1, R_2, P_0, P_1 $ and $P_2$. Each of these variables draws a smooth line through time. My question is, is there any ...
0
votes
2answers
119 views

Newton's Law of Cooling Example

A $200°F$ cup of tea is left in a $65°F$ room. At time $t=0$ the tea is cooling at $5°F$ per minute. Write an initial-value problem (differential equation with an initial condition) that models the ...
0
votes
2answers
35 views

Solve linear differential equation

So I have the following linear differential equation $$t\frac{dy}{dt}-3y=t^4$$ My first step was to divide through by $t$ to give $$\frac{dy}{dt}-3t^{-1}y=t^3$$ Then to find the integrating factor ...
0
votes
1answer
40 views

Euler and differentials

Did Euler have juxtaposition of $dx$ to $f'(x)$ to denote multiplication of a "very small quantity" to $f'(x)$ to obtain another "very small quantity" $dy$? This seems to imply that $\frac{dy}{dx}$ is ...
0
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0answers
26 views

NACA Airfoil: mapping from the camber axis back to cartesian

So how the NACA 4-digit airfoil is defined is it's a quartic thickness function, defined along a camber axis. The camber axis can either be a straight line, or a piecewise quadratic that has a peak at ...
0
votes
1answer
27 views

Problem in Identifying Homogeneous Differential equation

The following equation is Homogeneous (source:wolfram alpha), and has the answer $(x/y)+e^(x^3)=c$ as solved by putting $y=vx$. $$y dx - x dy + 3*x^2*y^2*e^(x^3) dx = 0$$ or $$(dy/dx) = (y + ...
8
votes
0answers
126 views

$\tau$ structure of the sixth Painlevé equation

I am studying the isomonodromic deformations theory, which leads in the case of a $\mathcal{C}_{0,4}$ Riemann surface to the sixth Painlevé equation. I read that this equation had a ...
1
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1answer
27 views

How's the chain rule applied?

When developing Lagrangian formalism, it is essential to set generalized coordinates: $ x_{i} = x_{i}(q_{j},t)$ where $t$ is time. $q$ is the generalized cooridnate we wish to use. During ...
-1
votes
1answer
21 views

Why $\Delta y \neq \cos((2.03)^2+1)-2-(\cos(2^2+1)-2)$?

On computing $\Delta y$ from $x=2$ to $x=2.03$: If $\Delta y = f(x+\Delta x) -f(x)$ and $y=\cos(x^2+1)-x$ why $\Delta y \neq \cos((2.03)^2+1)-2-(\cos(2^2+1)-2)$ ? Asumming $\Delta x=0.03$ and ...
0
votes
0answers
29 views

Integrate a volume form

$\omega$ is the volume form in $\mathbb{R}^n$ given by $\omega(v_1,\ldots,v_n) = \det([v_1\cdots v_n])$. Let $B$ be the closed unit ball in $\mathbb{R}^4$, given by $B=\{(x,y,z,w)\mid ...
7
votes
2answers
129 views

Solution to $y'=y^2-4$

I recognize this as a separable differential equation and receive the expression: $\frac{dy}{y^2-4}=dx$ The issue comes about when evaluating the left hand side integral: $\frac{dy}{y^2-4}$ I ...
1
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2answers
26 views

What is the proper DE for those questions?

A tank starts with 500 liters of water with 1 kg of salt dissolved in it. A salt and water mixture with concentration 0.1 kg/L is poured into the tank at a rate of 2 L/min. The mixture is drained at 4 ...
2
votes
2answers
70 views

Differential of a rotated f(x, y) surface

I often hit this problem : Consider a surface defined by the equation $z = f(x, y)$, the differentials of this function are $\frac{\partial f}{\partial x}\mathrm{d}x$ and $\frac{\partial ...
2
votes
1answer
53 views

A counterexample for a smoth version of Tietze extension theorem

Is there any function $f:F\subset \mathbb{R}^2\rightarrow \mathbb{R}$ with $F$ closed such that $f|F$ is differentiable in every accumulation point but there is no differentiable extension to the ...
0
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0answers
73 views

How to solve first order second degree differential equation?

I'm trying to solve a differential equation, which, upon expanding gives a first order second degree differential equation. Here, $R$ is the radius of the Earth, $\mu$ is the frictional constant. Both ...
1
vote
1answer
23 views

Approximating monotonically increasing differential equation

I am trying to make sense of the Appendix of the paper (Cooper, 1986). The following model is presented: $$\dot{(BX)}=\gamma_1BX \\ \dot{(BXB)}=\gamma_2(BX)B \\ \dot{B}=\gamma_3(BXB)$$ Without ...
4
votes
4answers
121 views

What does $d\log\left(\frac{y}{x}\right)$ mean mathematically?

I am used to seeing derivatives written as $$\frac{df}{dx}.$$ But my economics professor keeps using notation like $$ d\log\left(\frac{y}{x}\right)$$ and I have no idea what this means. What does ...
1
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0answers
83 views

nonlinear integro-differential equation

I'm working on a engineering problem and I need to solve this nasty differential. I gave it a go with Laplace transforms, but no luck. Any ideas? Note: a, b, c, and k are constants. ...
1
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2answers
41 views

Is this writing incorrect?

If we want to find $\frac{d}{dx}\cos x^2 $ then is this writing incorrect $\frac{d}{dx} \cos x^2= \frac{d}{dx^2}\cos x^2 \times \frac{d}{dx} x^2 $
0
votes
1answer
15 views

The determinant of f is not invertible when f is zero when the norm of the function is constant.

Let $f:U\subset \mathbb{R^n}\rightarrow \mathbb{R}^n$ differentiable on the open $U$. If $|f(x)|$ is constant, then $Df(a)$ is not invertible for every $a\in U$. How can I prove that?
5
votes
1answer
83 views

If $ds$ is not a differential form, can I make sense of its intuitive notation somehow?

I understand that a line element is not actually a differential form but a $1$-density. My question is: is the notation $ds^2 = dx^2 + dy^2$ formal in any way? Can it be interpreted as outer or tensor ...
5
votes
2answers
120 views

How is an infinitesimal $dx$ different from $\Delta x\,$? [duplicate]

When I learned calc, I was always taught $$\frac{df}{dx}= f'(x) = \lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{(x+h)-x}$$ But I have heard $dx$ is called an infinitesimal and I don't know what this ...
1
vote
2answers
26 views

Find solution to the differential equation

$\frac{dB}{dx}+2B=50$ $B(1) = 50$ I tried separating the variables but that didn't work, and without separating the variable I'm not sure what to do.