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

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How might I go about forming a general solution for the following differential equation?

$\frac{df}{dt}+t^kf=t^k$ where $k\in \mathbb{Z}$ I've solved for the equation previously where $k=2$ to get $f=\frac{-1}{t-t\ln t}+c$ but am not sure how I should go about solving this generally.
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2answers
25 views

ODE problem, now a days asked quite frequently (inhomogeneous)

How to solve $$\frac {ds}{dt}+s=|t|,~s(0)=1.$$ Because i have never seen before such a inhomogeneous ODE, where on right hand side there is modulus function. What i did to solve it, i broken up ...
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1answer
24 views

Let $X(t) = e^{r(T-t)}/S(t)$. Find the SDE of $X(t)$ provided that $S(t)$ satisfies the BSM model.

This is the last part to a 3 part question! I am nearly done going through the questions I had difficulty with while studying, again, anyone's help would be greatly appreciated! Let $X(t) = ...
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1answer
28 views

Difference for summation in maple

When calculate the differential respect to $x$ with Maple, I got this result. $$\frac{d^4x^\lambda}{dx^4} = \frac{x^\lambda\cdot \lambda\cdot(\lambda-1)\cdot(\lambda-2)\cdot(\lambda-3)}{x^4}$$ Is ...
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1answer
66 views

How would you integrate $1+\ln x$? [closed]

Could someone please show me how to integrate $1+\ln x$?
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2answers
34 views

If $h(x)=x \ln x$ , find $h'(x)$. [closed]

I'm not sure how to differentiate this function. I guess you use the chain rule, but I am not getting the correct answer. Could someone show me how it's done?
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1answer
31 views

ln integration (differential equations problem)

I'm trying to solve $$\frac{(\sqrt x + x)\,dy}{dx} = \sqrt y + y$$ I can separate the variables and get $$\frac {dy} {\sqrt y + y} = \frac{dx}{\sqrt x + x}$$ I know that integrating $$\frac ...
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2answers
23 views

Finding the error in the surface area of a cube. when length = 3, error= ${1\over 4} $

Find the approximate error in the surface area of a cube having an edge of length 3ft if an error of ${1 \over 4}$ in. is made in measuring an edge I have to do this by using differentials and ...
0
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2answers
24 views

Show that ellipses $5x^2 +6xy+5y^2=C$ are integral curves of the ODE: $(5x+3y)+(3x+5y)y'=0$

The second question asks: What are its solution curves? These questions were extracted from Ordinary Differential equations: Garrett Birkhoff. What I did: I differentiated the first equation (of the ...
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2answers
39 views

Find the $\Delta y$ of $f(x)={1 \over x^2}$; $x=2; \Delta x = 0.01$

Find the $\Delta y$ of $f(x)={1 \over x^2}$; $x=2; \Delta x = 0.01$ when $\Delta y = f(x+ \Delta x) - f(x)$ So here's what I did: $$\Delta y = f(x+ \Delta x) - f(x) \\ \Delta y = {1 \over (x+ ...
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0answers
20 views

Is there such a thing as a “partial differential”, brother to “total differential”?

I am familiar with total differentials in the form $$ f = f(x,y,z) $$ $$ df = \frac {\partial f} {\partial x} dx + \frac {\partial f} {\partial y} dy + \frac {\partial f} {\partial z} dz $$ however, ...
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3answers
45 views

Total derivative notation help

consider the function $$f = f(x(t),y(t))$$ I know that its total derivative wrt t is $$\frac {df}{dt} = \frac {\partial f} {\partial x} \frac {dx}{dt} + \frac {\partial f}{\partial y} \frac ...
0
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5answers
47 views

What is the differential of 2x + sin2x

I can't figure out how to differentiate 2x + sin 2x. I'm not sure if I should multiply the 2 + cos 2x by 2 . Basically I want to know what is the correct way to differentire 2x + sin 2x
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2answers
45 views

How to solve the differential equation $(3x^2y -x ^2)dx+dy = 0$

I want to solve the differential equation $$(3x^2y -x ^2)dx+dy = 0 \tag1 $$ $$\text{Here}\quad M(x,y) = 3x^2y -x ^2, \quad\text {and} \quad N(x,y) =1 $$ $${\partial M \over \partial y} = 3x^2\quad ...
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0answers
8 views

Limiting subdifferentials

For function in below Figure, there are three non-smooth points, at $x = 0$ and $x = \pm2.5$. My Question is that: Do we have Limiting subdifferentials at $x = \pm 2.5$? Definition of Limiting ...
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2answers
42 views

Application of differential calculus

I am having some trouble with this question in my grade 11 math textbook. We are learning applications in differential calculus. It is a question that can be solved many different ways but our teacher ...
3
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2answers
87 views

Why is $\cos(x)dx$ a differential form

I am trying to understand the concept of the differential A differential $d$ is a map that sends functions to 1-forms A preliminary example is $$d\sin(x) = \cos(x) dx$$ So the operator $d$ sends a ...
3
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2answers
73 views

How to differentiate it? [duplicate]

$f(x)=(x-1)(x-2)(x-3)(x-4)(x-5)$ How can I differentiate this function without expanding it to the polynomial form. Am I underestimating some theory of equation concept associated with it?(I know the ...
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1answer
18 views

Differential of the volume of a rectangular parallelepiped.

Suppose I have a rectangular parallelepiped with base $A$ and height $l$. In order to calculate the volume I could use a double integral and integrate over the intervals $[0,l]$ and $[0,A]$ the ...
6
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2answers
264 views

Counterexample of the almost-inverse of the Fundamnetal Theorem of Calculus(Lebesgue).

Can anyone give me a counterexample to the following statement: Suppose $F \colon [0,1] \to \mathbb{R}$ is continuous and differentiable almost everywhere, then $F(b)-F(a)=\int_a^b F'(t)\, ...
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1answer
16 views

Prove that the map $\pi$ is a submersion

I'm trying to solve the following problem: Let $\tilde M$ be the set of real matrices $3 \times 2$ of rank 2. $$ (u,v) = \begin{pmatrix} u_1 & v_1 \\ u_2 & v_2 \\ u_3 & v_3 ...
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0answers
16 views

Fitting nonlinear differential equations to correspond a predefined solution

When modeling temporal dynamics of a biological process I stumbled upon a set of differential equations having the following matrix form: ...
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0answers
30 views

Some Questions of the Differential

Let $X,Y$ normed vector spaces and let $f:X\rightarrow Y$ a differentiable function in each point $v\in\ U$, where $U\subset X$ is open. Then we have the first differential: ...
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0answers
59 views

Stability analysis for differential inclussion

I have a differential inclusion of the form $\dot{x}\in F(x)$, with $F(x)$ a set valued function that is also upper hemicontinuous. I want to analyze if a critical point of the system, $x^*$, such ...
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1answer
48 views

Give a volume form on $\mathbb{RP}^3$ [closed]

I was asked to determine a volume form on $\mathbb{RP}^3$. I would really appreciate any help. Thanks in advance.
0
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1answer
24 views

Non-linear first order ODE

I need help with the following riccati equation $$\ y' = Ay^2 + By + C $$ in the situation where discriminant < 0 It must be solved analytically with steps. Thanks
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0answers
15 views

Determine the analytic expression of this 2-form in a chart

I don't know how to do the following exercise. I would really appreciate if anyone knows how to solve it. $$\textbf{d}\alpha: \mathfrak{X}(M)\times \mathfrak{X}(M)\ni (X,Y)\longrightarrow ...
0
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1answer
28 views

Calculating coefficients in a differential equation

It's a pretty open-minded exercise I found online. It says, you're advising a social network company and they're trying to model an equation for $u(t)$, being this the amount of active users in the ...
0
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0answers
21 views

Proofing that Differential is surjective

there I had to find the Differential of $f: M->S$ where $M$ are $4x4 \text{ matrices}$ and $S$ are the $4x4$ symetric matrices and $f(A)=A^T*D*A$. $D$ is a diagonal matrix with entries ...
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0answers
17 views

General solution using power series method

I'm trying to get the general solution to the differential equation, $${\partial^2y(x) \over \partial x^2}+4x{\partial y(x) \over \partial x} + y(x) = 0$$ using the power series method, about the ...
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2answers
47 views

Deriving $\Delta z=\frac{\partial y}{\partial x}\Delta x+\frac{\partial f}{\partial y}\Delta y+\alpha\sqrt{\Delta x^2+\Delta y^2}$

I was reading a math book, which contained. "Let us consider a function $$z=f(x,y)$$ of two variables. If it has continuous partial derivatives, we can prove that its increment $$\Delta ...
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2answers
34 views

Confusing differential concept for $f(x)$ and $f(x,y)$.

Okay, so I was reading the concept of "Increment and Differential of a Function" from the book "Mathematics: Its Contents, Methods and Meaning". It says: Let us consider a function $y=f(x)$ which has ...
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1answer
111 views

Why is this a fake proof?

I am aware of the "definition" of the total differential as follows: $$\mathrm{d}f = \frac{\partial f}{\partial x} \mathrm{d}x + \frac{\partial f}{\partial y} \mathrm{d} y.$$ Now, assume we wished ...
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1answer
40 views

Differentiating function with norm

Let E be a vectorial space of finite dimension. $E^*=E\setminus \{0\}$. $$f:E^*\rightarrow E^*$$ $$x\rightarrow \frac{x}{\langle x,x\rangle}$$ Is this function $C^1$? How should I proceed? ...
0
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1answer
29 views

How to obtain Laplace transform of {f(t-a)U(t-b)}

$f(t)=g(t-10)U(t-15)-g(t-10)U(t-20)$ The above $f(t)$ contains terms of the form $g(t-a)U(t-b)$, where $a$ doesn't equal $b$. Describe the form that $L\{f(t-a)U(t-b)\}$ takes. [Hint: The formula for ...
1
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1answer
24 views

Unit step function differential equation

Suppose we model a physical phenomenon with a 2nd order linear differential equation: $a_2$(t)$y''$+$a_1$(t)$y'$+$a_0$(t)$y$=$f(t)$, where 't' stands for time. In choosing an appropriate driving ...
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0answers
15 views

Find the series solution to y''+xy'+y=0 when x0=0. Show y1 and y2 are fundamental set of solutions.

i have done some work to get the problem to look like this (not sure if its correct): 2a2 + a0 + Σ ((n+1)(n+2)a(n+2)+nan+an)xn=0 then making the coefficients =0 i get a2=-1/2a0 also getting ...
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0answers
26 views

Notation for Differential

What does the expression $Df_{\vec{x}_0}$ where $f:\mathbb{R}^2 \to \mathbb{R}^2$ and $\vec{x}_0 \in \mathbb{R}^2$ represent? How does this compare to $Df(\vec{x}_0)$?
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1answer
21 views

Double Differential Linear equation

An inhomogeneous linear second-order differential equation is given by: x''(t) + 2x'(t) + 5x(t) = 26e^(t) cos(t) - 26e^(2t) + t t ∈ R a) Find all the solutions ...
0
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1answer
41 views

total differential of product of scalar & vector functions

I've probably made mathematical mincemeat out of this but, suppose I have a product of scalar and vector functions, such as the momentum $\mathbf{p} = m \mathbf{v}$. To keep it reasonably simple but ...
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1answer
20 views

Check that F*(Xp) is a derivation at F(p)

To show linearity is simple but I am stuck on derivation
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1answer
42 views

Power Series $x^2y''+y=0$

Actually, I tried to solve (x^2)y''+y=0 power series, I cannot get the general soultion or the relation at least $(x^2)y''+y=0 $ $ \sum_{n=2}^\infty c_n n(n-1) x^n + \sum_{n=0}^\infty c_n x^n$ $ ...
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0answers
32 views

Differential Equations - Crossing a River

I am an independently studying student, and I am trying to solve the question in the attached link. Essentially, I need help solving Equation (3) in the below attached picture for the ratio ...
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2answers
79 views

Why is it legitimate to perform multiplication with differentials dx?

Why is it legitimate to perform multiplication with differentials $dx$? For instance, from the statement $dy = 5dx$ one derives $\frac{dy}{dx} = 5$. I learned $\frac{dy}{dx}$ as a notation to mean ...
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2answers
26 views

Differential Equation, linear or non-linear?

I am new to the area of solving differential equations, and I came across the following differential equation and was wondering whether it was linear or non-linear: $dy/dx= x^3 + y^3$ I would have ...
0
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1answer
21 views

Finding an integrating factor and solving the differential equation

I'm having trouble with this problem, and was wondering if someone could lead me in the right direction. Here is the question. Show that $x^ay^b$ is an integrating factor of the equation $$(b+1)x ...
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0answers
13 views

Termwise integration of Fourier Series PROBLEM

I need help with this problem, I keep solving it my own way but am not getting the same series as a result.
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2answers
31 views

obtaining easy differential equation solution

I passed by a question of a differential equation and i need help solving it thought its easy but am new with differential equations. Let S be the solution of the differential equation : $xy' -y= ...
0
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0answers
23 views

How to simplify a vector expression involving some differential operators

I want to know if it is possible to simplify the following two expressions: \begin{align*} \left(D_x V(x)\right)^{T}\nabla_x\mathrm{div}_x\,u(x),\\\nabla_x\left(\mathrm{Trace}\left[\left(D_x ...
0
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2answers
32 views

Proving Rules of Differentials

Show that for any two functions $f(x,y)$ and $g(x,y)$ we have $d(f+g)=df+dg$. Not too sure how to go about this. How would I set up the question?