Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [ordinary-differential-equations]

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

0
votes
1answer
11 views

Trapezoidal iteration method for solving differential equation

I am learning on how to use various numerical methods to approximate solutions to differential equations. We are using R, to actually iterate these functions, but I am having difficulties wrapping my ...
0
votes
0answers
14 views

Monodromies of complex differential equation

Consider the system of first order differential equations \begin{equation} \begin{cases} \frac{dY}{dz} = A Y\\ Y(1)= I, \end{cases} \end{equation} on $\mathbb{C}^*$ and assume that $z=0$ is an ...
0
votes
0answers
13 views

angularly and lineraly accelerating particle

Let's say there's this particle that moves with a unit of time. Let's say this particle has a linear velocity and acceleration ($v$ and $a$) and an angular velocity and acceleration ($v_\theta$ and $...
2
votes
0answers
39 views

Solve differential equation $-f'(x)= a_1 f(a_2 x+a_3)$ with $f(0)=1$.

How to solve the following differential equation \begin{align} -f'(x)= a_1 f(a_2 x+a_3) \end{align} where $f(0)=1$. I looked around I think this falls under the category of discrete delayed ...
0
votes
0answers
14 views

Sketch phase portrait for autonomous systems

I have thus far mostly (only) been dealing with phase portraits where you compute eigenvalues and eigenvectors and from there are able to sketch the portrait. Now I came across a different type of ...
0
votes
1answer
21 views

How can this differential Question about an RC circuit be solved?

The voltage, $5e^{-t} \cos(100\pi t)$, is applied to a circuit with a capacitance, $0.9\times 10^{-6} \mathrm{F}$, in series with a resistance, $13\times 10^3\mathrm{\Omega}$. Show that $\...
0
votes
1answer
24 views

Properties of eigenvalues from differential equation

I have the matrix equation $$ M'(t) = A(t) M(t)$$ with the initial condition $M(0) = I$, with $I$ the identity matrix, and where both $M$ and $A$ are $3\times 3$ matrices, $A(t)$ is real and can be ...
0
votes
0answers
28 views

How to rearrange second order differential equation

How can i accomplish this? The temperature, $T (=T(X))$, of a fin of perimeter P with cross-sectional area is given by $$ \frac{d^2 T}{dx^2 }- \frac{hP}{kA} T=0$$ where h is the convection heat ...
0
votes
2answers
36 views

Linearly Independent or Linearly Dependent

Classify the following statement as true or false. Give a short proof or a counter example in support of your answer. The functions $f(x)=\cos^2(x)$, $g(x)=\sin^2(x)$, $h(x)=\sec^2(x)$, and $k(x)=\...
0
votes
0answers
18 views

Existence of continnous function so that given two functions satisfies a differential equation

Do there exists continuous functions p,q on $(-a,a), a>0,$ such that the functions $y_1(x)=x^2\sin{x}$ and $y_2=1-cos{x}$ satisfies the equation $$y''+p(x)y'+q(x)y=0 $$ on this interval. I assumed ...
1
vote
1answer
36 views

Similar, but different curve to critically damped harmonic oscillator solution

I am looking for a curve with similar attributes to a critically damped harmonic oscillator, but is slightly different. As a reminder, the "classical" damped harmonic oscillator equation is: $$x(t) =...
2
votes
2answers
42 views

Solution to $ xy \ \frac {d^2y} {dx^2} + (x\ \frac {dy}{dx} - 2\ y) \frac {dy}{dx} = 0$

How can I find the general solution to this: $ xy \ \frac {d^2y} {dx^2} + (x\ \frac {dy}{dx} - 2\ y) \frac {dy}{dx} = 0$ I have learned various methods including: 1. integrating factor method 2. ...
0
votes
2answers
24 views

What's the proof for this formula for non-separable differential equations?

The formula $$y = e^{-H(x)}[\int{e^{-H(x)}*q(x)dx}+c]$$ is the one in question, where $H(x) = \int{p(x)}dx$, for equations of the form $$y' + p(x)y = q(x)$$ This is the equation given by my teacher, ...
2
votes
1answer
33 views

Numerical solution of ODE with Delta function

I want to model a dynamical system of the form $\frac{\text{d}x}{\text{d}t} = f(x)+nx\delta(\pi(t-0.2)). $ The problem is that I have a point source which is reoccurring at fixed time steps (say at ...
0
votes
0answers
27 views

ODE from solution

Given $f(x)$, for instance $f(x)=a+be^{cx}$ where $a, b, c \in \mathbb{R}$. Is there a theorem that establishes what nonlinear Autonomous ODE of any order has f(x) as a solution? or at least what ...
0
votes
0answers
58 views

Peano theorem — application to Cauchy problem

How do we prove existence of this Cauchy problem $$ \begin{cases} y'= f(x,y)= y \ln|y|\\ y(x_0)=y_0 \end{cases} $$ using Peano theorem? I try this: First, we have that $$ \dfrac{\partial f}{\...
0
votes
0answers
20 views

Non-Existence of the solution of a differential equation. [duplicate]

Show that the differential equation $|x'|+|x|+1=0$ has no solutions. Can we say that since $|x'|+|x|=-1$ and since LHS is positive then we cannot have any solution?
0
votes
0answers
25 views

The other solution of the Hermite equation

The Hermite equation is $$ y'' - 2x y' + 2 \nu y = 0 . $$ By using the Laplace method, people get a solution $$ y = A \int_C \frac{ e^{xt - t^2/4} }{t^{\nu+1}}dt , $$ where $C$, the path of ...
1
vote
1answer
38 views

Coupled differential equations into system of first-order equations implicitly

I am looking to solve the following equations numerically: $a x=\frac{d}{dt}\left(f(x,y,t)\frac{dy}{dt}\right),\quad b y=\frac{d}{dt}\left(g(x,y,t)\frac{dx}{dt}\right)$ For arbitrary functions $f$ ...
1
vote
1answer
29 views

Bernoulli’s equation problem

Find an explicit family of solutions for the Bernoulli’s equation $x^2y’+2xy=5y^4$ using a substitution. I worked out this problem but I’m not sure if I’m getting the right answer. If someone could ...
0
votes
3answers
44 views

How to solve this separable differential equation

This is the equation: $ y'\sin x + y\cos x = 0 $ condition $y(\pi/2)=4$ My solution: $dy \sin x = -y\cos x dx $ $\int dy/y = \int \cos x/\sin x dx $ $u =\sin x ,~~ du =\cos x dx$ $ - \ln y = \...
0
votes
0answers
23 views

Solving system of differential equations with unknown eigenvalues

I have 4 differential equations and a characteristic polynomial like $ λ ^4+ \frac{w^2*λ^2}{ɛ} - \frac{2kw^2}{ɛm}=0 $ where I denoted $ɛ$ as small deviation approximately zero, $m$: mass, $w$: angular ...
0
votes
1answer
84 views

ODE solutions - how un-differentiable can they be?

Suppose we have a first order Ordinary Differential Equation, $y^\prime(x)=f(x,y)$. On the face of it, it looks as though any solution $y$ should be differentiable throughout its domain, but this may ...
0
votes
0answers
20 views

Solution of a matrix differential system which verifies $\int_{0}^{+\infty} \lVert A(t) \rVert dt < \infty$ admit a finite limit over $+\infty$

Let be $A : \mathbb{R}_{+} \to M_n(\mathbb{R})$ a continuous function such that: \begin{equation*} \int_0^{+\infty} \lVert A(t) \rVert dt < \infty \end{equation*} Let be $u$ a solution to the ...
-1
votes
2answers
37 views

Does $f(t,y(t))=4t\sqrt{y(t)}$ satisfy the Lipschitz Condition condition?

$f(t,y(t))=4t\sqrt{y(t)}$. Does $f$ satisfy the Lipschitz Condition condition? In other words how do I check if $$|f(t, y_1) - f(t, y_2)| = |4t\sqrt{y_1} - 4t\sqrt{y_2}| \le C |y_1- y_2|$$ holds. ...
-2
votes
1answer
37 views

Solving the following ODE $f′+f=1$ with the usage of power series? [on hold]

How can I solve the following differential equation: $$f′+f=1$$ with the usage of power series? (https://i.stack.imgur.com/g6AOv.jpg)
0
votes
0answers
63 views

Existance and unicity of Cauchy problem

i have the following Osgood Lemma: let $f(x,y)$ a function such as $|f(x,y_1)-f(x,y_2)| \leq h(|y_2 - y_1|)$ for all $(x,y_1)$ and $(x,y_2)$ in an opena $\Omega \subset \mathbb{R}^2$. We suppose that ...
1
vote
1answer
31 views

What is the final answer of $m\dot{v}=mg-kv^2$?

I tried solving this equation using several different methods but I never got to an answer. I tried using partial integration and at last I got $m/\sqrt{mg}\ln mg-kv^2 = t+c$ I even tried using ...
2
votes
2answers
48 views

Exact Differential Equation Integrating Factor

Finding an integrating factor can be a genuine mathematical art. However, certain differential forms can remind us of differentiation techniques that may aid in the solution of the equation at ...
0
votes
0answers
21 views

Finding $x_0$ and $t_0$ such that a sharper version of the existence and uniqueness theorem applies?

I'm not too sure what a 'sharper' version of the existence and uniqueness theorem refers to here. Suppose I'm given: $$f(x,t) = |t|sin(x)$$ I know that for the theorem to apply, I show that $f(x,t)...
0
votes
2answers
37 views

How to find the general solution of this equation?

$$y'=(xy'+y)y^3$$ I don't know how to approach this problem with the two $y'$.
0
votes
1answer
28 views

Drug Concentration with half life with constant dosage?

I am stuck on a problem whereby, a drug has a half life of 36 hours, but every 24 hours 100g is added including at the start. So Q(0)=100g, I know how to do half life calculations but am struggling to ...
0
votes
1answer
33 views

Finding a relation between functions according to known constraints

I am solving a problem on geodesics with ideas from General Relativity and got stuck with one step. The simplified version is the following: With notations $$\dot{x}\equiv \frac{dx}{dt}, \quad \...
1
vote
1answer
40 views

Visualise the bifurcation diagram

Can anyone help me visualise the bifurcation diagram that would be produced by $\dot x = (x−μ)(1+μ−x^2)$ and $\dot x = (μ^2−1)(μ−2)−x$ I know for 2. there is only one equilibrium point so ...
0
votes
0answers
20 views

Tenenbaum and Pollard ordinary differential equations

Exercise 10, question 14. I cannot work this out after hours of trying. The suggested integrating factor does not seem to make the equation exact. Any help most appreciated.
2
votes
1answer
85 views

Solving a PDE in $1$D

I want to solve the below equation $\partial_t f(x,t)=D \partial_x^2 f(x,t)+\partial_x f(x,t)$ with initial and boundary conditions: $f(x,0)=\delta(x-x_0)$ $D\partial_x f(x,t)+f(x,t)=0$ at $x=L$ ...
0
votes
2answers
49 views

Solve Radiation Total Energy Equation

So I have a system that I assumed is lumped (perfect conduction) with heat transfer only through radiation with source $A$. The equation is as follows $$C \frac{dT}{dt} = A - BT^4$$ $A = 13.876$ , ...
1
vote
1answer
47 views

To determine a constant in an ODE [on hold]

Let $w(r)$ be a function of $r$, we have the following ODE: $$r^{n-1}w'+\frac{1}{2}r^nw=a$$ for a constant $a$. Assume the equation holds for all positive integer $n$. The book claims that if ...
1
vote
0answers
34 views

Power series ordinary differential equations

I have the equation: $y'' - (\frac2x + x^2)y' + y = 0$ I need to use power series of the form $y = \sum_{n=0}^\infty a_n(x-5)^{n}$ and coefficients up to n=4 Derivatives of y: $y' = \sum_{n=1}^\...
0
votes
1answer
37 views

Differential Equation Involving Multiplication of Infinite Series?

I am trying to solve the differential equation $y'' + e^x y = 0$, where $y(0) = 1, y'(0) = 0$. In my class, we are told to use the substitution $y = \sum_{j=0} ^{\infty} a_{j}x^{j}$. However, when it ...
3
votes
0answers
47 views

Characterizing a certain kind of bifurcation

I'm dealing with the following ODE, $$ \left(\frac{dr}{d\lambda}\right)^2 = \left[1 + \frac{(C - 1)M}{(2r - 3M)}\right]\left[1 - \left(1-\frac{2M}{r}\right)\left(\frac{D}{r}\right)^2\right] \equiv f(...
1
vote
0answers
24 views

Proving bound on the solution of a first order linear ODE.

Given, $d’ + fd = -E$, where $|E(t)|\leq e^{-\delta t}$, $\delta >0$. $0<c_0<f(t)<C_0$ for all time $t>0$. Prove that $|d|<e^{-\delta t}, \forall\ t>t_0,$ for some $t_0$ ...
0
votes
1answer
37 views

Airy's equation

$I(y)= \int_ \gamma \mathbb{e^{zy-\frac{z^3}{3}}}dz$. This integral is taken along the contour composed of $\mathbb{R_{+}}$ and $ \mathbb{e^\frac{2\pi}{3}}$*$\mathbb{R_{+}}$. Prove that this integral ...
2
votes
0answers
54 views

Stuck on an integral while solving an ODE

I was solving $$\left(1+3e^{y/x}\right)\,\mathrm dx+3e^{y/x}\left(1-\dfrac xy\right)\,\mathrm dy=0\\ \implies \dfrac{\mathrm dy}{\mathrm dx}= \dfrac{1+3e^{y/x}}{3e^{y/x}\left(\dfrac xy-1\right)}$$ ...
1
vote
0answers
19 views

Solving the matrix ODE $M\ddot x + C\dot x + Kx = \sin(\omega t)e_i$

Consider the following ODE, where $x$ is a vector, $M, C, K$ real square matrices and $e_i$ a basis vector (for example $[1\ 0\ \dots 0]^\top$).: $M\ddot x + C\dot x + Kx = \sin(\omega t)e_i$ with $...
1
vote
1answer
26 views

Please what has gone wrong with these transformations?

Let the equality $$x^2-y^4=6$$ define $y$ as a function of $x,$ where both variables are real. The problem I set out to solve was to find the second derivative of $y$ with regard to $x,$ and I found ...
-1
votes
0answers
40 views

Asymptotic stability of a nonautonemous and non linear system

I have been asked to decide if solutions to the system $ \dddot{x} - \ddot{x} + cos(tx + \dot{x}) = 0 $ are asymptotically stable. Tried linearization but think it's wrong. Is it true that i should ...
0
votes
0answers
34 views

Draw phase plane in $(x,y)$-plane given system of differential equations in polar coordinates

Given $ \begin{cases} \frac{dr}{dt} = r(1-r),\\ \frac{d\varphi}{dt} = \sin^2\varphi + (1-r)^3, \end{cases}$ I want to draw the phase plane in the $(x,y)$-coordinate system with $x=r\cos(\varphi)$ ...
6
votes
1answer
133 views
+50

Two fluids flowing perpendicular in thermal contact with a Wall [Help to mathematically model]

I will try to describe briefly how I am modelling the problem. (Please bear with the length). The governing equation describing temperature for a block at steady state is $$\nabla^2 T = 0$$ where $\...
0
votes
1answer
28 views

Determine the flow of the differential equation $\dot{y}=Ay$

Determine the flow of the differential equation $\dot{y}=Ay$, where $A=\Big(\begin{matrix} 2&1\\0&2 \end{matrix}\Big)$ The solution to the differential equation would be $y(t)=e^{(t-t_0)A}...