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Questions tagged [differential-equations]

Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.

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0answers
15 views

System of Differential Equation with initial conditions

I am trying to solve the following system of equations: $du/dt = -u / tu$ $dw/dt = (x-w)/tw - u*w$ In the above system, $tu,x,tw$ are all constants. The question is using initial conditions ($u=...
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3answers
30 views

Finding a geometric series for $\frac{1}{(1 + x)^2}$

I'm asked to find a geometric series for $f(x) = \frac{1}{(1 + x)^2}$, I integrate it first and I get $F(x) = \int{\frac{1}{(1 + x)^2}dx} = \frac{1}{-2(1 - (-x))}$ Since $\frac1{1-x} = \sum_{n=0}^\...
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2answers
31 views

Solve for $x$ and $y$ where $x^{\prime} = x + y + t$ and $y^{\prime} = 2x - t$. Given $x = 0$, $y = 1$ at $t = 1$. [on hold]

Given the equations \begin{align} x' &= x + y + t \\ y' &= 2 x - t \end{align} with $x(1) = 0$ and $y(1) = 1$, how can solutions be obtained? I tried solving it but I did not get any ...
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1answer
23 views

Solution for O.D.E on $\mathbb{S}^n$,

Given $a \in \mathbb{S}^n$, let be $X(q) = a - \big<a,q\big>q$ vector field on $\mathbb{S}^n$. Give $q_0 \in \mathbb{S}^n$, find an solution for $ \alpha'(t) = X(\alpha(t))$.
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4answers
35 views

Non-linear second order ODE

I have to solve $$ y^{''}(x)+(y^{'}(x))^2=y'(x). $$ Using $ y'(x)=z $ I can write $\int \frac{1}{z-z^2}dz=\int dx $. So: $$\frac{1}{z(1-z)}=\frac{A}{z}+\frac{B}{1-z}$$ leads to \begin{align} \int ...
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0answers
35 views

Differential Equations (ODE). Book.

I read "Ordinal Differential Equations" by V.I. Arnold (in russian). In 1st lesson he told about differential 1-forms and explained form's ingration. Also he told about monodromy. In the next lesson ...
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0answers
12 views

Laplacian recursive definition

I am reading about Laplacian in hyper spherical coordinates "So, the Laplacian in hyperspherical coordinates consists of the radial portion and the angular part. The transformation from $(r, \Omega_{...
0
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1answer
25 views

Find the general solution (GS) of the DE

i) $(1 + t)y^{\prime} = ty+t+t^2$ ii) $y^{\prime} + \cot(t)y = 2cost$ Not sure how to start...
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2answers
36 views

Find a solution of $t\frac{dy}{dt}=t^2−t$ and determine a function $y(t)$ that passes through the given coordinates $(t, y)$

Find a solution of $t\frac{dy}{dt}=t^2−t$ that passes through the points: i) $(0, 1)$ ii) $(0, 0)$ iii) $(1/2, 1/2)$ iv) $(2, 1/4)$ SOS: I don't know where to start and my professor is no help.
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1answer
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Find the restricted space within function and line

I have the following question: Let $f(x)=x + sin(x)^2$ defined for every $-\pi\le x \le\pi$. Find the entire restricted space within the $f(x)$ and the line $y = x$. I drew the intersecting ...
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0answers
23 views

Constructing a differential equation involving local isometries

I bet there must be some research about this topic I'm not able to find. Suppose I have a smooth surface patch $\sigma_0$ I want to construct a local isometry $\sigma_{\infty}$ with some specific ...
0
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1answer
32 views

Do we consider differential equations a pure math?

I want to understand the field of Differential Equations because I want to make it my future major...I am so confused because I am in pure math section and I feel like lights on algebra and topology ...
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1answer
68 views

Solving the Cauchy problem: $y' = \frac{xy}{(x-1)^2}, y(2) = 1 $

I want to study this differential equation, that is to justify the unique maximal solution and determine the interval where the solution holds (if it exists). We have: $$\left\{\begin{matrix} y' =\...
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2answers
31 views

MATLAB Error Implementing Fourth Order Adams-Moulton Method

Trying to implement the fourth order AM method in MATLAB using fourth order RK to get the first four starting values. Code is as follows: ...
1
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1answer
25 views

Undetermined coefficient in recurrence relation

I am given $3x^2(x+2)y''+7xy'-2y=0, x \geq 0$. I am asked to solve this differential equation with a series solution around $x=0$. Note, however that $x=0$ is a regular singular point since: $$ P(x) =...
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3answers
149 views

Solve Inverse Laplace Transform Using Input Integral Theorem

Problem Using the input integral principle below $$ \mathscr{L} \left[ \int_{0}^{t} f(u)du \right] (s) = \frac{1}{s} \mathscr{L} \left[ f(t) \right] (s), \ s > c $$ Find $ \mathscr{L}^{-1} \left[...
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2answers
27 views

Separable differential equation that results in a complicated integral

I am having trouble solving this differential equation: $$y\left(3x+\frac{6x^2\sin^2(\frac x2)}{x-\sin x}\right)\,\mathrm dx=\frac{\sqrt x\,\mathrm dy}{(x-\sin x)^{\frac32}}.$$ I know it is separable ...
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2answers
32 views

Nonlinear differential equation existence and uniqueness theorem

What is the nonlinear differential equation existence and uniqueness theorem? I have two solutions to a nonlinear first-order differential equation that both satisfy the same initial value and satisfy ...
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0answers
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Custom loss function for YOLO algorithm with tensorflow or keras backend [on hold]

Each grid in my YOLO output vector(7*7*3) has depth of only 3 units. First one indicates probability of object in that grid(have only one object to localize), and other two predicts coordinates of ...
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1answer
23 views

Linear Matrix differential equation

Let $A,B$ be non-singular matrices of dimension $n\times n$. Is there a way to solve the differential equation $$ f(x)Bx=A\nabla_x f(x)? $$ I've looked in many places and it doesn't seem to be ...
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2answers
76 views

Solve $u'=u \ \cos(x)$

\begin{cases}u'=u \ \cos(x) \\ u(0)=1 \end{cases} $$u'=u \ \cos(x)$$ $$\frac{du}{dx}=u \ \cos(x)$$ $$\frac{du}{u}=\cos(x) \ dx$$ $$\int \frac{du}{u}=\int \cos(x) \ dx$$ $$\ln \lvert u \lvert=\sin(...
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0answers
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Stuck in solving the Hermite DE?

As a part of solving the Schrödinger equation1 for quantum Harmonic oscillator: \begin{equation}\tag{1} \left(-\frac{d^2}{d\tilde{x}^2} + \tilde{x}^2\right) \tilde{\psi}(\tilde{x}) = \tilde{E} \...
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0answers
21 views

How to solve a coupled, first-order, non-linear equation with first derivatives of multiple functions.

I am interested in solving a differential equation of the form $$\left(\frac{dx}{dr}\right)^{\!2}=F(x,y)+G(x)\left(\frac{dy}{dr}\right)^{\!2}$$ The reason most of my tricks don't work are that $F$ ...
0
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1answer
31 views

First order non-linear ODE with error function

I have to solve $ y'(x)=-2xy(x)+ey^2(x) $. Using $ z=y^{-1}$ and $-z^{'}=\frac{y^{'}}{y^{2}}$ i arrive to prove that $ z^{'}=-2xz+e $, but when i apply the variation of constants method i obtain $ ...
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0answers
18 views

Invertibility of derivation on $AP(\mathbb{R},\mathbb{R}^n)$

I have following question based on this paper: https://boundaryvalueproblems.springeropen.com/track/pdf/10.1186/s13661-016-0576-9 The author defines the function spaces $AP(\mathbb{R},\mathbb{R}^n)=...
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0answers
24 views

How to compute the actual current of Poisson equation

Consider Poisson equation $\nabla \cdot (\sigma(x)\nabla u)=0$ in a domain $D$, where $\sigma(x)$ is the spatially dependent conductivity. On the boundary we have $n$ electrodes (Dirichlet BC $u=\text{...
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2answers
51 views

Solution to $u_x + y u_y = -u$ with $u(0, y) = \cos y$ [duplicate]

I am studying for my PDEs test and I want to make sure I can solve this type of equations. I used the method of characteristics. $$\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{b}{a} = y$$ Integrating I ...
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1answer
42 views

Finding equilibrium and bifurcation points for first order ODE's

I'm trying to find the equilibrium and bifurcation points for $$\frac{d}{dx}=rx+xe^{-x}.$$ I started by noticing that $x=0$ is an equilibrium point for all $r$. Also, I then set the RHS equal to zero ...
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0answers
35 views

How do I compute the eigenfunctions of an operator that contains another operator?

Given the operator $A = (X\frac{d}{dx}+2)$, where $X$ is a linear operator, how can I find the eigenfunction of $A$ corresponding to a zero eigenvalue? In general, this is just a matter of solving ...
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0answers
36 views
+50

Matrix Regression for linear ODE system

Background I have the following homogeneous ODE system as an Initial Value Problem: $$ y'=A\cdot y\quad\wedge\quad y(0)=y_0 $$ where $y\in\mathbb{R}^{N\times 1}$ is the unknown vector and $A\in\...
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1answer
37 views

Symmetries and invariants.

I would like someone to give me an example of application of the following paragraph from an introductory book on Lie groups theory: The paragraph states the following: Symmetry transformations, if ...
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0answers
28 views

Eigenvalue of $(\partial_\tau-\frac{1}{2}\triangledown^2_x+1-\xi(\vec{x},\tau))\Psi(\vec{x},\tau)=\lambda\Psi(\vec{x},\tau)$ [on hold]

Originally, I want to evaluate the path integral $\int D\Psi^{\dagger}D\Psi e^{-S}$, where S is the imaginary time action $S=\Psi^{\dagger}(\partial_\tau-\frac{1}{2}\triangledown^2_x+1-\xi(\vec{x},\...
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1answer
37 views

theorema Egregium and coeficcients of the second fundamental form

The theorema Egregium says that Gaussian curvature $K$ of a regular surface $S$ is invariant under local isometries. We have a local description of the Gaussian curvature as follows $$K = \dfrac{eg-f^...
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0answers
30 views

Harvesting equation bifurcation

Could anyone give me some pointers on how to make a bifurcation diagram of a two parameter ODE of a harvesting model. It's $$ \dot{x} = ax\left(1 - \frac{x}{b}\right)- \frac{x^2}{1 + x^2}. $$ If I ...
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0answers
87 views

How to solve differential equation $f'(x) = f(x) - f(x-1) $ with $f(0)= 0$ [duplicate]

How can I solve the ODE: $f'(x) = f(x) - f(x-1)$ where $f(0)= 0$, and $x \geq 0$ Note: I am trying to solve a problem where it came up and I haven't done DE for years and thus not sure where to ...
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1answer
26 views

Initial value/differential equations problem

The problem in question: The population of mosquitoes in a certain area increases at a rate proportional to the current population, and in the absence of other factors, the population doubles each ...
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2answers
46 views

One Parameter family of soultions for this Riccati's Equation

Find a one parameter family of solutions of the Riccati's equation $$\frac{dy}{dx}=-\frac4{x^2}-\frac{y}{x}+y^2$$ So I am trying to follow a sort of template I was given: Let $y=y_1+u$, where $...
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1answer
29 views

Distance of the point from the curve

Original question The solution of the equation $$ \frac{dy}{dx}= \frac{3x-4y-1}{3x-4y-3}$$ passes through the origin. The distance of it from the point (-1,1) is ___. *My attempt * Found the ...
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1answer
26 views

Step in Exact Differential Equations

I'm going through this "Elementary Differential Equations and Boundary Value Problems" book, by Boyce and DiPrima, and I have a little trouble understanding a passage in the Exact Equations ...
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3answers
37 views

ODE Non linear Non separable

$$ \frac{dy} {dx} = \frac{3x-4y-2}{3x-4y-3}$$ I don't know how to solve this. Searched about the equation to know about the non separable one. I know about the separable one. Any hint will be helpful....
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Identifying ODE types for solving by hand and when to use computers instead

So this questions relates to my specific ODE but also ODEs in general. I am a big fan of solving ODEs by hand, but I also know when to give up and use, say, Mathematica to solve it for me. Having ...
2
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2answers
46 views

Show $\frac{du}{dx}=|u| \ (x\in\mathbb{R}$) has solutions $u=Ae^x$

Show the following ODE has solutions of the form $u=Ae^x$: $$\frac{du}{dx}=|u|, \ x\in\mathbb{R}.$$ My attempt: I first considered the case where $u>0$. So, \begin{align} \frac{du}{dx}&=u \\ \...
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0answers
36 views

Show that a function is not differentiable using epsilon delta

show that the function z= y+ix is not differentiable at any point using the definition of epsilon delta. So I know that the definition of epilson delta is: f is differentiable at a with derivative f′(...
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0answers
16 views

Changing of independent variables

I have a set of differential equations with 5 dependent variables and 2 independent variables (r, t). One of the 5 dependent variable (R) is monotonically decreasing with the independent variable t. I ...
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0answers
40 views

Intervals of Existence of the solution to $x^2 y' = (1 - x^2)(1 + y^2)$

I'm curious about the intervals of existence of the DE in question. It is solvable via separation of variables, upon which one receives $$ y = \tan\left(-\frac1x - x + C\right) $$ for arbitrary ...
3
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3answers
235 views

Solution to this Differential Equation $f''(x)=f(x)f'(x)$ needed

I came up with this differential equation and I don't know how to solve it. $$f''(x)=f(x)f'(x)$$ I attempted to solve it several times, but they were all fruitless. Wolfram Alpha says that the ...
1
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2answers
52 views

Can a non-constant solution of DE intersect an equilibrium solution.

Consider the autonomous DE; $$\frac{dy}{dx}=\frac{y^3-9y}{e^y}$$ Find the equillibrium solutions and classify each as stable, unstable or semi-stable. Can a non-constant solution of this ...
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2answers
46 views

Find the solution of this differential equation

Find the function y solution of $$(x+1)(x^2+1)y'=2x^2+x$$ Initial conditions : $y = 1$ when $x=0 $ I tried to simplify the equation first by breaking down the fraction in two sides. so that I will ...
2
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2answers
41 views

solve this 1st order linear equation

$$(x+3)^2\frac{dy}{dx}=6-12y-4xy=6-y(12+4x)$$ a. Write it in standard form. $$\frac{dy}{dx}+\frac{12+4x}{(x+3)^2}y=\frac6{(x+3)^2}$$ b. What is the integrating factor? $$\frac{12+4x}{(x+3)^2} = \...
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0answers
22 views

Understand some points of Method of Characteristic and the solution of the Wave Equation.

I started my first pde course. I don't have much experience with this yet. I found an interesting question: Solution to one-dimensional Wave Equation with Method of Characteristics, but I didn't ...