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

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ODEs - Newton's Second Law

So there's this question on a past exam I'm trying to work through, and its solution: Is it possible to make it so that v'=-v^2? That was my first approach and then to integrate directly to get the ...
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1answer
26 views

Fourth Order Homogeneous Ordinary Differential Equation With Double Complex Conjugate Roots (2.10-14)

This is actually a problem in algebra as shall be seen. I need to find the general solution for the following differential equation: $$y''''+8y''+16y=0$$ The characteristic equation for this is: ...
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1answer
74 views

Basic Initial Value Problem

Given the initial value problem $$x''+4x=0, \qquad x(0)=1, x'(0)=4$$ (a) Find the matrix $A$ for which $\begin{bmatrix}x'\\x''\end{bmatrix} = A \begin{bmatrix}x\\x'\end{bmatrix}$. (b) Find ...
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0answers
31 views

First order non linear differential equation solution with maximal interval of existence

I have some doubts about my way of resolving a Cauchy problem (with a differential equation as a separable equation). Here is the resolution, some questions at the bottom: $$ y(t)y'(t)=1, y(0)=-2 $$ ...
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31 views

Branching Processes

I would like to have some hints about the following problem: Imagine a descendant tree. We assume that each species can give birth to a new specie in a constant rate $\lambda$. (This rate is called ...
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1answer
70 views

Help for Integral and evaluating - Eikonal equation

Hy guys I'm reading a paper of "Finding Exact Solutions to the Two- Dimensional Eikonal Equation" - E.D. Moskalensky. link for the paper: ...
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2answers
48 views

Exact solution for ODE: $yy' + y + f(x) = 0$

Is there is exactly solution for ODE in the form: $yy'+y+f(x)=0$. Thanks. If there is no such solution for general $f$, does it ease the problem if $f(x)=Ax+B$ for some constants $A$ and $B$? ...
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0answers
13 views

Positivity of compartments in epidemiological model

Given the following dynamical model (system of ODEs): \begin{array} $ \frac{dA}{dt}=\Lambda-\mu A-\beta(C+D+E+F)\frac{A}{N}-\tau(B+D)\frac{A}{N} \\ ...
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3answers
78 views

Differential equation $\frac{y}{y+x}dx = dy$

I am trying to solve the problem $(x+y)u_x + yu_y = 0$ with the condition $u(x,x) = x$. I assume that $u_xdx + u_ydy = 0$ Now I am wondering, how do we solve $\frac{y}{y+x}dx = dy$?
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1answer
14 views

Linear independence of three simple functions (2.9-22)

Why is the following set of three functions linearly independent on an interval $I$ with $x>0$ if $k_2$ can take on any value besides zero and still hold true for a zero sum or that $k_1=k_3$ which ...
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37 views

How to solve differential equation $f(y^2)\sqrt{y^2+y'^2}-\frac{f(y^2)y'^2}{\sqrt{y^2+y'^2}}=C$

I have the following differential equation I need to solve: $$f(y^2)\sqrt{y^2+y'^2}-\displaystyle\frac{f(y^2)y'^2}{\sqrt{y^2+y'^2}}=C,$$ where $y=y(x), \;y'=y'(x)=\displaystyle\frac{dy}{dx}, ...
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1answer
34 views

Confusion in writing Complementary function of ODE

I have the differential equation as $( D^{2} + a )y = 0$ , where D is differential operator I have roots of D as $\pm$ $\sqrt{a}i$ I am having confusiin in this part which is that Should i ...
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1answer
44 views

Population differential equation

So my friend and I got this question for our differential equations class and we cannot figure it out. Consider a population N (t) that is changing according to the following rules: the per capita ...
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2answers
38 views

using the substitution $ z = x- y,$ solve $y^\prime = y-x-1 + (x-y+2)^{-1}$ [closed]

Can you please help me with the question bellow? using the substitution $z = x- y,$ solve $$y^\prime = y-x-1 + (x-y+2)^{-1}$$
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1answer
33 views

Gronwall's Lemma type problem

I have a function $X(t)\geq 0$, with initial condition $X(0)=X_0\geq 0$ and constants $\alpha < 0$, $\beta > 0$ and $\gamma <0$ such that $$\frac{d}{dt} X(t)^2 \leq \alpha X(t)^2 + \beta ...
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1answer
46 views

Solve the differential equation: $ (y\cos(x)+2xe^y)+(\sin(x)+x^2e^y-1)y'. $

Solve the differential equation: $$ (y\cos(x)+2xe^y)+(\sin(x)+x^2e^y-1)y'=0. $$ I can rewrite the equation as $$ \frac{d}{dx}(y\sin(x)+x^2e^y-y)=0 $$ to get $$ y\sin(x)+x^2e^y-y=C $$ but how do I go ...
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1answer
52 views

System of differential equations, pure imaginary eigenvalues, show that the trajectory is an ellipse.

I am stuck at the last part of a proof. When you have the system of equations: $x'=Ax$ $$ A=\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ ...
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0answers
42 views

Differential equation of order 2

Solve the differential equation $$y''(x)-4y'(x)+8y(x)=10e^x\cos(x)$$ I am not able to solve this particular differential equation. Please help. I know the answer is: $$y(x) = ...
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2answers
53 views

First Order Differential Equation $y' \cos x +y=\sec x+\tan x$

I'm stuck on a seemingly straight forward problem as follows: $$\cos x \frac{dy}{dx}+y=\sec x+\tan x$$ I have rearranged the equation to be: $$\frac{dy}{dx}+\sec x \cdot y=(\sec x+\tan x)\sec x$$ ...
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27 views

Application of the Existence and uniqueness theorem

Given the initial value value problem $$y'=\frac{10xy^{0.4}}{3}$$ $$y(0)=-1$$ has a unique solution on some open interval that contains $x_{o}$. Find a solution and determine the largest open ...
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What is the geometrical interpretation of differential equation?

I was wondering what was the geometrical interpretation on differential equations, is there one ?
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2answers
31 views

To find an extremal of a given functional

I have to find extremal of following : $\int_0^1 [(y')^2 + 12 xy] dx$ with $y(0) = 0$ and $y(1) = 1$. I applied the Euler's equation $\frac{\partial F}{\partial y} - \frac{d}{dx}(\frac{\partial ...
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52 views

How to solve Energy Balance equation by numerical method

Good Day I am new to heat transfer technique please give me some suggestion on solving energy balance equation $$a \frac{\partial T_p}{\partial t}=\frac{\partial}{\partial x}\left(b\frac{\partial ...
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1answer
45 views

2nd order differential equation with missing y'

I have the following 2nd order differential equation: $$y'' + p(x) y =0, \tag{1}$$ where $p(x)$ involves only first order of $x$, for example, $p(x)=ax+b$. Any suggestion how to obtain or guess a ...
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28 views

book suggestion on manifolds

I've to learn differential equations on Manifolds. Can any one please suggest some books/lecture notes for differential equations on Manifolds ?
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1answer
49 views

Differential Equation $y'=\frac{1}{x^2-1}, y(0)=0$, am I missing something?

I'm encountering a singularity of sorts when working out this D.E. It seems like a very straight forward problem and, assuming I'm going wrong, I'm wondering where. The problem: Solve the initial ...
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2answers
33 views

Integrating Factor - Exact Equation problem.

I have stumbled with a problem I can't seem to solve. $$(x^2 - y ^2)dx - 5xy dy = 0$$ We know that $$u(x,y) = \frac{1}{(x M + y N)}$$ if the equation is HDE (Which it is..I believe). Excuse my ...
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2answers
35 views

Autonomous equation having $\frac{t^2}{1+t}$ as a solution

Find an autonomous equation having $\displaystyle\frac{t^2}{1+t}$ as a solution. So the desired function $f$ should depend only on $x$, if I'm not wrong in the form $x'=f(x)$, that means the goal ...
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40 views

Fourier sine series of $f = \cos x$

Let $f:(0,\pi) \to \mathbb{R}$ defined by $x \mapsto \cos x $ Show that the Fourier sine series of (odd extension) is given by $$\sum\limits_{n=2}^\infty \frac{2n(1+(-1)^n)}{\pi(n^2-1)}$$ So far, ...
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0answers
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Stability of non-autonomous stochastic differential equation

I'm looking for a good reference or insight to under what conditions can I prove stability (or instability) for the following general n-dimensional non-autonomous stochastic differential equation: ...
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0answers
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Determining linear independence of three simple functions for a third order ODE. (2.9-7)

This is a very similar post to one previous by me but I felt that not all questions were satisfactorily answered. But I am sincerely grateful to those who tried. I would like a sharp independent eye ...
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5answers
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Separable diff. eqn: $(1+x^2)y' = x^2y^2, x > 0$

I have been given a step-by-step answer which I just cannot understand or follow. $\begin{eqnarray} &(1+x^2)y' &= x^2y^2 + y\cdot1 \\ \iff& \frac{1}{y^2} &= \frac{x^2}{1+x^2} ...
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1answer
29 views

To find solution of differential equation

Find the continous solutionof $$\frac{dy}{dx} +y = G (x),\qquad x \geq 0,\quad y (0) = 2 $$ where $$G (x) = \begin{cases} 3 & \text{when }x\in [0, \pi/2) \\ \cos x & \text{when $x\ge ...
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1answer
22 views

How to solve this first-order nonlinear ordinary differential equation?

Obtain the solution to the DE $$\dfrac{dy}{dx} = \dfrac{1+y^2}{1+x^2}$$ (A) $\dfrac{Cx}{1-Cx}$ (B) $\dfrac{Cx}{1+Cx}$ (C) $\dfrac{C-x}{1-Cx}$ (D) $\dfrac{1-Cx}{x+C}$ (E) ...
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24 views

Differential equation with $(\frac{dx}{dt})^2$

I have had very little training in differential equations, so this might be a stupid question. Is it okay to solve something like $t = x^2({dx\over dt} )^2$, by doing: $t(dt)^2=t^2(dx)^2 \rightarrow ...
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42 views

High dimensional Differential Evolution

I want to minimize a cost function with Differential Evolution (DE) algorithm and I have 55 unknown parameters as an input for DE algorithm. Therefore, the DE should search in high-dimensional space ...
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1answer
29 views

string displacement function by d'Alembert's formula

Consider an infinite string stretched taut on $x$ axis from $-\infty$ to $\infty$ . Let the string be drawn aside into a curve $y=f(x)$ and released, and assume that its subsequent motion is described ...
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First Order Differential Equations - length of the arc joining two points on it

What curve lying above the x axis has the property that the length of the arc joining any two points on it is proportional to the area under that arc?
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2answers
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Determining linear independence of three simple functions for a third order ODE. (2.9-6)

I would like a sharp independent eye other than my own to review my work here. I have a few questions I would like answered. Did I actually answer/solve all parts of this problem? Determinants of ...
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3answers
58 views

Finding a particular solution to a differential equation [closed]

what is the particular solution for the following differential equation? $$D^3 (D^2+D+1)(D^2+1)(D^2-3D+2)y=x^3+\cos\left(\frac{\sqrt{3}}2x \right)+xe^{2x}+\cos(x)$$ I tried Undetermined Coefficients ...
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Singular solution to ODE and singularities

I'm a bit confused about those two, i'll try to explain. I get an ODE like $$ \frac {y \cdot dy }{\sqrt{y^2+1} } + \frac {x \cdot dx}{ \sqrt {x^2 +1}} = 0 $$ What I'm not sure about : 1) at y=0 the ...
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1answer
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Solve the pde $(x^2-y^2-yz)p+ (x^2-y^2-zx)q=z(x-y)$

I'm solving this by Lagrange's method. Lagrange's auxiliary equation is: $\frac{dx}{x^2-y^2-yz}=\frac {dy}{x^2-y^2-zx}= \frac{dz}{z(x-y)}$ From the first two ratio and the last ratio: ...
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1answer
37 views

ODE Separable Equation

Let $y = Φ(x)$ be a solution to $y' = y(5-y)(8-y)$ subject to $y(0) = 7$. Determine $\lim_{x \to ∞} Φ(x)$. Workings: I'm thinking I have to solve the differential equation. $y' = y(5-y)(8-y) dy$ ...
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1answer
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Verify that each given function is a solution of the given partial differential equation

$\alpha^2u_{xx}$ = $u_t$; $u_1(x,t)= e^{-\alpha^2t}\sin x$ I took the derivate of u1 and then took the second derivate and plugged it into $\alpha^2u_{xx}$ = $u_t$ as $u_{xx}$ but it is looking ...
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1answer
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Solving the Helmholtz equation

If I wanted to figure out how to do a simulation with the Helmholtz equation, how would I do it? Or, what kinds of techniques would I have to learn in order to figure it out? Background: 1st year ...
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Is there another way of solving this differential equation?

$y'+y\tan x=\dfrac{1}{\cos x}$ I found the integrating factor to be $e^{-\ln\cos x}$, but then I get into having to integrate $\dfrac{-\ln\cos x}{\cos x}$ , which seems to be quite a mess. Is there ...
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1answer
29 views

First order differential equation problem

Suppose we have $$ \frac{dy}{dx} +f(x)y = r(x) $$ and it has two solutions $y_1(x)$ and $y_2(x)$ then how to prove that solution of differential equation $$ \frac{dy}{dx} +f(x)y = 2r(x) $$ Will be ...
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42 views

What's wrong with my solution for the following differential equation?

I have the following DE: $$y'+2xy-xy^4=0$$ It's a Bernoulli equation, so I converted it to: $v'-6xv=-3x$ and the integrating factor being $e^{-3x^2}$ after doing the necessary steps, I find myself ...
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2answers
28 views

First order DE, need help

I am trying to solve this equation by inspection: $$(xy-y)dx+(x^2-2x+y)dy=0$$ Hints would be very helpful.. Thanks
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59 views

Solving Laplace equation in polar coordinates

I have some assignments to do and I don't even know where to start. The notes in the course aren't too good, so I didn't understand too much from them. Given $$ \Omega = \{(x, y) \in \mathbb{R}^2 , ...