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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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21 views

ODE becomes unsolveable (unstable) when transformed from second order to third order.

I am working on a coupled system from fluid dynamics that is highly dynamic and often unstable. I have reproduced the results of a famous paper that first studied the given system and came up with ...
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3answers
26 views

I need help finding the general solution to the differential equation $y''(t)+7y'(t)=-14$

What I've tried: I have the inhomogeneous differential equation: $$y''(t)+7y'(t)=-14$$ I find the particular solution to be on the form $$kt$$ by inserting the particular solution in the equation ...
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1answer
20 views

How to solve these simultaneous differential equations

I want to solve the following equations for $x_1$, $x_2$, and $x_3$ $$ K_0\, x_1(t) + K_1\, \frac{dx_1(t)}{dt} + M_0\, \frac{dx_2(t)}{dt} + x_3(t) =A_0\,\sin (\alpha_0 t) \tag{1} $$ $$ M_0\, \frac{...
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0answers
7 views

Finite difference methods for optimal control problems

I am currently analyzing data that, a-posteriori, can be viewed as the values of an euler discretization with fixed step size 1 of an optimal control problem. Can somebody recommend an introductory ...
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1answer
25 views

Find the particular solution of, $y=Ce^{-2x}+De^{-3x}+\cos(x)+\sin(x)$

Where, $y = 1$, $dy/dx = 0$ when $x = 0$. I've tried using simultaneous equations but keep getting $0$ as an answer for both constants, not sure how else to proceed.
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1answer
24 views

Transform a non-linear differential equation into a linear equation

Following the work of Yaoji Lu - 1967 (here's a link to the full paper) I got stuck at the step when the author transform a non-linear differential equation into a linear equation (eq. 3.9 pag. 19). ...
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0answers
14 views

transform the equation into Bessel equation

$$\frac{d}{dx} \left(x^a \frac{dy}{dx}\right)+bx^h y=0$$ Show that equation can be transformed into a Bessel equation in terms of $t$ and $u$ by transforming both independent and dependent variables ...
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0answers
19 views

Confusion about exact DEs

If I understand the definition of an exact de correctly, if $M(x,y)dx+N(x,y)dy=0 $ and $M_y=N_x$, then $$f(x,y)=\int (M(x,y)dx + g(y)$$ and $$g'(y)=\int M(x,y)dx-N(x,y)$$ 1) Is my interpretation ...
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0answers
21 views

Poincare-Bendixson Theorem to prove existence of a nontrivial periodic solution

Use the Poincare-Bendixson Theorem to prove the existence of a nontrivial periodic solution of the following DE: $z'' + (z^2 + (z')^4 - 2)z = 0$
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2answers
62 views

Does $y'=|y|^a$ have any global solutions?

Assume the differential equation $$ y'=|y|^a $$ My intuition tells me that since it involves an absolute value, there might not be any solutions defined everywhere, except for the case $a=0$, where $...
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0answers
27 views

Ward identities for 1D integrals

Some integrals of type $$ I(a)=\int_{-\infty}^\infty dx~ e^{i(x^3/3 + a x)},$$ can be solved by realizing that they satisfy a differential equation like $$ \left(a - d^2/da^2\right)I(a)=0,$$ ...
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1answer
23 views

Euler-Ansatz for hom. ODEs with constant coefficients

Given is an hom. ODE with constant constant coefficients: $A_0y(x)+A_1y'(x)+A_2y''(x) + \dots + A_ny^{(n)}=0 \tag{1}$ Now it's clear to me that the solution space $y\in\mathbb L$ is a vector space. ...
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1answer
24 views

solving a DE with Frobenius method

I'm trying to solve: $xy''+(5-x)y'-y=0$ using frobenius method, which I'm keep getting a wrong answer. can someone explain how to solve this problem? Also, are there any sources I can refer to? I ...
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0answers
13 views

Help with ODE solving using RK method in Matlab

I have the following ode $i\dot{x_n}(t)=-x_{n+1}(t)-x_{n-1}(t)+g~ x_n(t)$ with I.C. $\phi(0)=-2\cos(2.5)$ and $g=1$. where $\dot{x_n}=\frac{dx_n}{dt}$, $n$ are discrete points in space, and $x$ is ...
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2answers
94 views

How to solve $\dot{x}=|x|$

How to solve this differential equation $\dot{x}=|x|$?
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0answers
14 views

Draw phase diagram of $x'' + V(x) = 0$ conservative system, with $V(x) = \omega^2 \cos{x} - \alpha x$

I want to draw the phase diagram of $x'' + V(x) = 0$, with $V(x) = \omega^2 \cos{x} - \alpha x$, with all constants greater than $0$. I write this second order differential equation as a system in ...
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0answers
22 views

Convert pair of second order DEQs to four first order equations

enter image description here how to do it please help, I know how to solve a simple one variable nth order equation
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1answer
39 views

Find the largest interval where the Initial Value Problem $y'(t)=t+sin(y(t))$ with $y(2)=1$ has a unique solution. Using Picard-Lindelof Theorem.

Consider the initial value problem (IVP) \begin{cases} y'(t)= t + \sin(y(t)), \\ y(2) = 1. \\ \end{cases} Find the largest interval $\mathcal{I}\subset \mathbb{R}$ containing $t_0=2$ so that the ...
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1answer
38 views

Limit of a solution to a differential equation is a steady state.

Suppose we have an initial value problem $$\dot{x}=f(x),$$ $$x(0)=x_0$$ where $f\in C^1(E)$ for some open $E$. Moreover, suppose we have a solution $x(t)$ such that $$\underset{t\rightarrow\infty}{\...
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1answer
18 views

Solving differential equation that includes an (extra) unknown function

I am trying to find a solution for $C_a(t)$ in the below differential equation: $$ V \cdot \frac {dC_a(t)} {dt}=R(t)-Q_v \cdot C_a(t) \\ $$ $ \text{Where} \\ \qquad V \text{ is the volume of the test ...
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0answers
21 views

The second order accuracy of TR-BDF2 method

The following method is called the TR-BDF2: $$U^* = U^n+\frac k4(f(U^n)+f(U^{*}))$$ $$U^{n+1}= \frac 13(4U^* - U^n+kf(U^{n+1})).$$ Apparently, this one-step method is second-order accurate while being ...
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0answers
27 views

attracting set dynamical systems

There are some definitions about attracting set, but one of them is confusing for me. First, definitions: $$ \begin{array}{l} 1) \textrm{flow:} \enspace \phi_t({\bf x}) \\ 2) \textrm{for a subset} \...
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1answer
23 views

If $f$ and $g$ both satisfy Cauchy Riemann equations at point $z$, prove $f+g$ and $fg$ satisfy the Cauchy Riemann equations

So if $f=u+iv$, and $g=m+in$. Trying to prove $fg$ follows the Cauchy Riemann equations: I got $fg=(mu-nv)+i(nu+mv)$, letting $a=mu-nv$ and letting $b=nu+mv$, I differentiated $a$ and $b$, by $x$ and $...
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0answers
19 views

The roots of Bessel functions

In Abramowitz Stegun 1972 there is an inequality (9.5.2) for roots of Bessel functions and their derivatives (n is positive): $$ n \leq j^{'}_{n,1} <y^{}_{n,1} < y^{'}_{n,1} < j^{}_{n,1} <...
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1answer
42 views

Find the general solution of the system of ODEs [on hold]

$$\begin{cases}x'_t=-\dfrac54x+\dfrac34y+\dfrac2{1+e^t}\\y'_t=\dfrac34x-\dfrac54y\end{cases}$$ I get the eigenvalues $-1/2$ and $-2$, but the number of eigenvectors for $-2$ is $2$, I don't know why.
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2answers
53 views

Unable to solve differential… $yy'=y+1$

The differential equation is: $$yy'=y+1$$I've been trying to solve this problem all evening, but I cannot figure it out, and none of the online calculators show me the steps on how to find the ...
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0answers
18 views

For what values of the parameter $c$ the difference equation have periodic (2-cycle) solutions

I have difference equation: $y(n+1)=y(n)^2+c$ At what value of parametr $c$ the equation have a 2-cycle solution? In this example $f(y)=y^2+c$ To solve the problem, I wanted to solve this equation: ...
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0answers
16 views

Monotonicity of Bessel functions [on hold]

Prove or disprove: n-th Bessel function (where n is integer) is monotonic on interval [0,n]. I have tried to prove it from basic properties about Bessel functions but i didn't managed to do this. The ...
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0answers
39 views

Picard theorem for $u' = \sqrt{\lvert u^2 -1 \rvert}$ if we know $ u(\pi / 2)= 0$

Problem: can we apply Picard theorem for $$u' = \sqrt{\lvert u^2 -1 \rvert}$$ if $$ u(\pi / 2)= 0$$ [$u$ is a function of a variable $x$ so $u = u(x)$] My attempt: Well, what I need to know is ...
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1answer
18 views

Transition from one difference equation to another

I have initial equation: $y(n+1)=y^2(n)+C$, $C\leq\frac{1}4$ How I can show that if $C=\frac{a}2+\frac{a^2}4$ then by using replacement $y(n)=\alpha x(n)+\beta$ I will get the equation $x(n+1)=...
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0answers
21 views

Finding a specific bump function

I am trying to smooth the edges of a linear function $f(x)=-1/C * x +1$, where $C \in \mathbb{R}$ is a constant, in such a way, that my ultimate function is $ \equiv 1$ in a neighbourhood of zero, and ...
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1answer
43 views

Global Attractor Existence

How do I know if a system of differential equations has a global attractor? I am an undergraduate physics student. I am trying to understand what is the domain of applicability of Convergent Cross ...
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2answers
50 views

What is the intuition behind the method of undetermined coefficients?

Our teacher has recently begun teaching second-order differential equations and the methods used for solving them. The method which we are taught to solve linear differential equations is currently ...
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1answer
29 views

Find the equation of the curve $y''=2x+1$

I am stuck on the last step of the following equation: If $y''=2x+1$ and there is a stationary point at $(3,2)$ find the equation of the curve. So far I have the following: If $y''=2x+1$ $y'=x^2+x+...
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1answer
20 views

Constructing a differential equation with a bifurcation

I am trying to construct a differential equation $x' = f_a(x)$ where the number of equilibrium solutions depends on $a$ in this fashion: if $a<0$, no equilibrium; if $a=0$, one equilibrium; if $a&...
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0answers
23 views

Write a differential equation representing the rate of change of grams of chocolate/ minute

I have one cup of hot milk and I stir in melted chocolate that is 24g/cup at a rate of 1cup/min. So that my cup does not overflow, I pour the mixed drink out at the same rate. I was given the answer ...
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1answer
33 views

Understanding logistic growth

The population growth can be modelled by the function ${dP\over dt}=rP(1-{P\over k})$ and $P$ will go from $P_0$ (the initial value) to $k$. But, I am trying to understand the behaviour of the term $...
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0answers
17 views

Integral Transforms and Partial DEs: Heat Equation of a rod

Consider a rod of length $l$ whose lateral surface is insulated. Let its initial temperature be zero, then maintain one of its end faces as at zero temperature and the other at $u_0$. A. Determine the ...
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1answer
52 views

How to Solve this Differential Calculus Problem

If the equation of the normal line to the curve $y = ax + b/x$ at the point $(2,7)$ is $y+ 2x = 11$, find the value of $a$ and $b$. Given that this normal line meets the curve again at $P$, find the ...
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0answers
31 views

Bifurcation in 1D with two parameters

I'm trying to do bifurcation analysis of the system $x'=-x((x-2)^{2} - \mu^{2} -9)((x+2)^2 - \mu^2 -9)+ \epsilon$ When $\epsilon = 0$ this is easy because we have some nice circles. But I'm stuck as ...
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0answers
18 views

Help solving very complex first order ODEs using ode45 - MATLAB - movement of water

I am trying to solve this complex set of first order ODEs using ode45 on MATLAB. They describe the change in volume of water in the inner layer ($X_I$) and outer layer ($X_O$) of the Venus Flytrap's ...
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1answer
22 views

Equation of the normal to a curve [on hold]

I am struggling to find the equation of the normal to the line: $$y = \frac{1}{x} - \frac{3}{x^2} - \frac{4}{x^3} + \frac{7}{4}$$ at $(-2,1)$. Any ideas would be appreciated. I believe I need to ...
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0answers
35 views

How to define and solve this PDE? Is it Burger's equation?

Can someone help me with defining this PDE? What possible analytical or numerical solution can be applied here? Thanks in advance!
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0answers
12 views

Arbitrary constants of 2nd order linear ODE with negative discriminant are complex conjugates?

I'm in a mathematical modeling course. My professor was showing us how to solve initial value problems regarding 2nd order linear ODEs with a negative discriminant. He asserted that the arbitrary ...
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0answers
25 views

Solution of complex differential equation involving $2$ variables

Solve the differential equation $\displaystyle \frac{dy}{dx} = \sqrt{\frac{1}{2}+\int^{-\sin^4 \theta}_{\cos^4 \theta}\frac{\sqrt{f(\phi)}d\phi}{\sqrt{f(\theta)}+\sqrt{f(\cos 2 \theta-\phi)}}},...
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0answers
18 views

Is the unique solution of autonomous ODE infinitely differentiable

Consider the ODE \begin{align} y'&=f(y)\\ y(0)&=y_0 \end{align} where $f$ is Lipschitz and, if it matters, always positive. When restricted on an interval $[0,a]$, is the unique solution of ...
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0answers
27 views

Multiplying Integrals with Different Bounds

The question I have is two integrals mutliplied with different bounds $$\alpha_0(x)=\int_x^1\exp\left(\frac{-\varphi\lambda^2}2\right)d\lambda\cdot\left(\int_0^1\exp\left(\frac{-\varphi\lambda^2}2\...
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0answers
16 views

Prove: $f(t) \leq K_1 e^{K_2(t-a)}+\frac{\varepsilon}{K_2} e^{K_2(t-a)-1}$

Let $f$ be a non-negative function that satisfies $$f(t) \leq K_1 + \varepsilon(t-a)+K_2 \int_a^t f(s)ds$$ for $a\leq t \leq b, \quad K_1,K_2,\varepsilon \in \mathbb{R}$. Prove: $$f(t) \...
2
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0answers
37 views

How to prove that there is no periodic orbit in this autonomous system?(ODE)

Consider the autonomous system below with the parameters $a\neq 0$ and $mn \neq 0$: $$\begin{cases} \displaystyle\frac{dx}{dt} &=-y+mxy+ny^2\\ \displaystyle\frac{dy}{dt} &=x+ax^2 \end{cases}$...
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1answer
21 views

Showing that a second order differential equation has unique bounded solution

I'm trying to show that $x''+bx'+x=\cos(t)$ has a unique bounded solution, for $b<0$, $b\in\mathbb{R}$. I believe I understand that it has a unique solution as all coefficients and the non-...