Questions tagged [differential]

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

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find the general solution of the given differential equation y'' - 2y' + 5y = xe^x + sin(2x)xe^x [closed]

I was trying to find the general solution to this question but have been unable to do so. y'' - 2y' + 5y = xe^x + sin(2x)xe^x
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1answer
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general solution: absolute value [closed]

Hey I have a problem including absolute values without an initial condition. I looked all over my textbook and the internet and cannot solve it. $$\frac{dy}{dt} = |y|^{1/2}$$ I did positive: $$2y^{1/2}...
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2answers
33 views

Gradient Descent on differential equation

I have a differential equation of the form $$\frac{d}{dx}f=f^2$$ I want to find a root of the second derivative of $f$, in order to maximize the derivative $df/dx$. I could of course simply solve the ...
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1answer
20 views

How to think about this simple differential calculus expression? [duplicate]

In the course of doing differential equation substitutions, I often come across the expression $\frac{d}{dt}\frac{dy}{dx}$, where $y$ is a function of $x$, and $x$ itself is a function of $t$. For ...
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2answers
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Winding number of a curve (not complex analysis)

I am asked to calculate the winding number of an ellipse (it's clearly 1 but I need to calculate it) I tried two different aproaches but none seems to work. I would like to know why none of them work (...
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Solutions to 2nd order elliptic pde with mixed boundary conditions

Let $\Omega$ be a simply-connected smooth domain in $\mathbb{R}^2$, let $\Gamma_1\cup\Gamma_2=\partial\Omega$ and $\Gamma_1\cap\Gamma_2=\{p_1,p_2\}$. Consider following problem \begin{cases} Lu=f, &...
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28 views

Estimates of the remainder in Taylor's Theorem (Wikipedia)

Read the first two lines of this section I think it is sufficient to say that $f$ is $(k+1)$-times differentiable in an interval $I$ containing $a$ with $$q \leq |f^{(k+1)}(x)| \leq Q$$ for all $x \in ...
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1answer
33 views

Find general solution of $dx/dy + x/y = −y^5 x^9$ using Bernoulli [closed]

How to find general solution of $dx/dy + x/y = −y^5 x^9$ using Bernoulli? A first-order Bernoulli ODE has the form of $y' + p(x)y = q(x)y^n$.
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29 views

How to define the differential on a manifold/How to proove a function is differentiable over a manifold?

How to define the differential on a manifold ? How to prove a function is differentiable over a manifold ? I've seen some definitions but can't get to a proper understanding of them. To what i've ...
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Integration (differential equation)

Does anyone know how to solve: $$(2x^2-xy)dx-(y^2+xy)dy=0$$ I solved that $$\frac{dy}{dx} = \frac{(2x^2-xy)}{(y^2+xy)}$$ Then, I substituted $vx$ for $y$ and found that $$-\frac{dx}{x}=\frac{(v^2+v)dv}...
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What method is appropriate for the following differential equation?

Differential Equation I am stuck at this equation what method is appropriate and how I should solve it? $$\left(x e^y-1\right)dy= -\left(e^{x+y}+e^y y\right)dx$$ Equation
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1answer
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Gateux differential question

I am going over the book "Optimization by Vector Space Methods" by Luenberger, and I found a statement that I'm not sure I understand. I am attaching the relevant page and I drew a box ...
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6answers
68 views

When doing separable differential equations, why does Integrating a constant give a variable? ie. $\int -3 d𝑦 = -3y$

I get that it is abit handwavy at this point in my math journey, but why does integrating a constant like this: $\int -3 dy$ result in -3y? I get that when integrating a variable like y gives $\frac {...
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1answer
34 views

Differential form equation conversion to derivative equation

Suppose $x=x(t)$. Let $dx$ and $dt$ be 1-forms. Does there exist a rigorous theorem for converting $dx=f(t)dt$ into the derivative ("fraction") expression $\frac{dx}{dt}=f(t)$? Is this ...
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1answer
15 views

Outside Temperature- First-Order, Linear, Initial Value Differential Equation

I am trying to solve a first-order initial value differential equation. Here is the equation: Is this separable? I feel like you could solve it using an integrating factor where P(x)=k and Q(x) is ...
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1answer
57 views

Separating the variables possible?

Given the differential equation $$ \frac{dy}{dx} = y+x$$ I am told this differential equation is separable. Meaning I need to rewrite the RHS into a product of two variables depending on y and x. I've ...
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1answer
37 views

Unable to solve this differentiation

$$\mathbf{v}=d\mathbf{r}/dt=R( -\sin(\theta)d\theta/dt, \cos(\theta)d\theta/dt)=R\omega(-\sin(\theta), \cos(\theta))$$ Now, it’s differentiation equals in textbook $$\mathbf{a}=d\mathbf{v}/dt= R (-d\...
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1answer
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Confusion in chain rule of differentiation [closed]

Let sin(wt ) be the one. Let z = wt Then , we find dz / dt = w . Then , let y = sin z I am having confusion to why dy/dz will be - cos z and not only - cos . Since , in dz /dt , t gets cut right . Why ...
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1answer
46 views

How do i prove f(0) = 0 (Calculus problem) [closed]

I need some help to get started with this question. $$f(x) + \ln(1+f(x)) = \frac{\sin x}{1+x}$$ Show $f(0) = 0$ and use that to find $f'(0)$. Where do i start to show $f(0) = 0$? $f(x)$ inside $\ln(1+...
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2answers
51 views

How to determine if this Differential Equation has a singular solution?

I found this Differential Equation $y'^2=16x^2$ and the text says it has a singular solution... first they make an implicity derivation with respect to $y'$ then they have $2y'=0$. Therefore $y=...
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Is this definition of the gradient using the exterior derivative consistent with calculus 1?

I'm trying to get a better grasp on what the exterior derivative means on my own and I'm trying to connect the language of forms to my pre-existing knowledge. I came along the following formula for ...
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1answer
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Showing the smoothness of the function with two variables

In the last step of the proof, I have to show that $f:(x,y)\mapsto\sqrt{1-x^2-y^2}$ is a smooth mapping, where $x,y\in\mathbb{R}$. From direct computation with the usage of induction, I know that this ...
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If $f(x,y)$ is $C^{\infty}$ in $x$ and $y$ respectively, can we say that $f$ is $C^{\infty}$ in $x$ and $y$ simultaneously?

$f$ is a function from $\mathbb{R}^2\to \mathbb{R}$. I feel like this can't be true but I need to come up with some counterexample, but I'm not sure how. Can you please help?
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2answers
28 views

Show that differential equation has integrating factor $\eta (x, y) = \nu (x^2 + y^2)$ and find it

I have differential equation $$x + x^4 + 2x^2 y^2 + y^4 + yy' = 0$$ The problem is to find integrating factor $\eta (x, y) = \nu (x^2 + y^2)$ for such big equation :( Does anybody have any ideas?
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1answer
32 views

Determinant of a differential operator of the action variations

I have to compute the determinant of a differential operator that comes from an action variation: $\frac{\delta^2S}{\delta q(\tau)\delta q(\tau)}=-M\frac{\partial^2}{\partial \tau^2}+V''(q)$. It is ...
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2answers
37 views

solve the Bernoulli equation xy' - y = xy^2

Solve the Bernoulli equation $xy' - y = xy^2$. I started with diving both sides by $x$, and ended up with $y' - \frac{y}{x} = y^2$. Then, I divided both sides by $y^2$ and got $\frac{y'}{y^2} - \frac{...
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Are Differential equations without “$x$” considered separable?

For example: Do we consider all differential equations of the form $y'(x)=f(y)$ separable? why/why not?
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1answer
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Difference between geometrical and graphical representations.

My books has a sub-topic called geometric meaning of differential equations. It mentions that slope of tangent at every point in 2d space is from given differential equation. According to it, without ...
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Differential Equations Laplace Transformation

I need help with this question. It has to be solved using laplace transformation Solve the following initial value problem using Laplace transform $$y''+2y'+y=te^{-t},\quad y(0)=1,y'(0)=-2,$$
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Arguments to show that $\lim_{\iiint d\tau\to 0}\frac{\iint d\vec{\sigma}\times\vec{V}}{\iiint d\tau}=\nabla\times\vec{V}$

So, I was solving the problem 1.10.6 from "George B Arfken, Hans J Weber - Mathematical Methods For Physicists- Sixth edition" and the problem was to show that: $\lim_{\iiint d\tau\to 0}\...
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2answers
123 views

Is it inuitively correct to say that $\lim_{n \to \infty} \frac{x}{n} = \mathrm{d}x$?

We know that from the Riemann sum that $\Delta x = \mathrm{d}x$ where $n \to \infty$ and $\Delta x = \dfrac{b - a}{n}$. If, however, $x$ represents some length of interval, can we also say that $\...
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Solving Exact Differential Equations Short Cut/Second method

I remember reading something about another way to solve an exact differential equation by ignoring particular terms in the equation and then determining if what you have left is the derivative of ...
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1answer
43 views

Does dx in dA=dx*dy represent change, or is it a notation that denotes an infinitesimal arbitrary length? Change vs. an arbitrary physical length

I'm not sure how to explain this, but I have a gap in my understanding of infinitesimals/differentials. I've so far had calc 1 and 2, and have been taught that dy/dx represents a slope, which ...
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1answer
21 views

Get $\frac{d^2y}{dx^2}$ from $2x(\frac{dy}{dx}) = \frac{dy}{dt}$

I stumbled upon this on a "worked solutions manual" But I dont quite understand the part I selected in red. [![This is the working out they show][1]][1] How can $ \frac{dy}{dt}$ be converted ...
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2answers
26 views

Tangent parabolas

Given that $y^2=x$ and $x^2=y$ are both parabolas, what is the value of $c$ if $y^2=x$ and $x^2+c=y$ are both tangent? What I have tried: $$\begin{cases} y^2=x\\ x^2+c= y \end{cases}$$ Taking the ...
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25 views

Vector image under the differential of the $\dfrac{1} {z}$ function

I have a confusion with the following problem: Show that the image vectors of the vectors $(1,0)$ and $(0,1)$ under the differential of $\dfrac{1}{z}$ are orthogonal for all $z\in \mathbb{C^*=C\...
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Differential Equations Curve families problem

How can i find the curve family Whose distance to the origin(0,0) is slope times ordinate at that point ( slope * y-coordinate) in the plane.
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28 views

Elementary course on exterior differential calculus

I am looking for a bibliographic source for a course on external differential calculus that is accessible at a very "low level" in university; first or second year of university in physics ...
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1answer
26 views

The hessian of max [closed]

I was trying to understand the Hessian matrix. I asked this question what is the the hessian of the max and the min of two given functions. My question is the following : what is the exact expression ...
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1answer
79 views

Using Differentials (not partial derivatives) to prove that d𝜃/dx = -sin(𝜃)/r [duplicate]

I am trying to prove the parts of each component of the inverse matrix in the attached image. I have tried using differentials and then solving for the other components. (I'd like to solve it this way)...
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51 views

Laplace Transforms and Convolutions

enter image description hereHello, Im working with convolutions and using step functions to solve differential equations. Can someone please show me how to derive the second superposition formula from ...
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1answer
65 views

Showing that $d\theta(\partial\Psi/\partial\theta)=1$

In Bachman's A Geometric Approach to Differential Forms, second edition, the following is exercise 8.4. Often we parametrize surfaces in $\mathbb{R}^3$ by starting with cylindrical or spherical ...
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3answers
54 views

Why the differential of a function is a linear transformation?

I try to find some applications of the differential, that weird thing that is a linear transformation, what is the reason of linearity? And how this linearity works, I mean, there's something ...
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19 views

what is the meaning of each term in the two-dimensional wave equation?

$u_{t t}=\alpha^{2} u_{x x}+\beta^{2} u_{z z}+\left(\alpha^{2}-\beta^{2}\right) w_{x z}$ $w_{t t}=\beta^{2} w_{x x}+\alpha^{2} w_{z z}+\left(\alpha^{2}-\beta^{2}\right) u_{x z}$ were $u$ and $w$ are ...
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8 views

What does it mean agreement in norm mean in differential equations?

I am taking a course in numerical analysis in physical systems. Trying to understand the next definition: Absorbing boundary condition: Suppose $ u$ solves the well-posed differential equation $$ Lu=f,...
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2answers
81 views

Complex chain rule

Suppose $V_1$ and $V_2$ are open sets in $\mathbb{C}^m$ and $\mathbb{C}^n$, repsectively, $f:V_1\mapsto V_2$ and $g:V_2\mapsto \mathbb{C}^k$ are $\mathbb{R}-$ differentiable. We know that the real ...
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1answer
28 views

Sturm separation and comparison theorem result:

Let $u"+p_1(t)u=0$ and $v"+p_2(t)v=0$ with $p_2(t)>p_1(t)$ in $(a,b)$. Suppose that $u(a)=v(a)=0$ and $u'(a)=v'(a)=x>0$. Show that there exists $\epsilon>0$ such that $v(t)>u(t)$ in $(a,a+\...
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1answer
43 views

Understanding the definition and notation of the second order differential

We're reaching the end of a differential calculus course I'm taking (distance learning), and I'm realizing that I don't fully understand the objects I'm manipulating. In particular, I'm not quite ...
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1answer
57 views

Differential Equation but by trial and error solution

Original question : Find $f(x)$, $x\geq 0$, such that $$f'(x) = \frac{\sqrt{f(2x)-1}}{\sqrt{2}}$$ My work : $\begin{align} f'(x) &= \frac{\sqrt{f(2x)-1}}{\sqrt{2}}\\ f(2x) &= 2(f'(x))^2 + 1\...
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1answer
60 views

How did they came up with this simplification of given differential equation? (kepler's first law proof)

So for the proof of Kepler's first law they make a simplification of a differential equation, but I don't get how they come up with that simplification. It goes like this $$2\frac{dr}{dt}\frac{d\theta}...

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