All Questions
Tagged with applications ordinary-differential-equations
117
questions
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Solving ODE system with less equations
In sensitivity analysis, there is a set of equations called sensitivity equations. They're obtained by differentiating your initial IVP with respect to the parameters. For example:
If your IVP is:
$\...
- 101
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0
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58
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How to formally justify fudge factor in this difference equation solution?
In Exercise $11$ from Section $3.3$ of Differential Equations With Boundary Value Problems by Polking, Boggess, and Arnold, we first develop the difference equation $P[n + 1] = (1 + \frac{I}{m})P[n],\ ...
- 1,922
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3
answers
53
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How to untangle the ODE $\frac{dx}{dt} = c + \frac{px}{l_0 + pt}$? [closed]
In working on this problem, I came up with the following differential equation:
$$
\frac{dx}{dt} = c + \frac{px}{l_0 + pt}
$$
where $x$ is the dependent variable, $t$ the independent, and all others ...
- 4,450
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1
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41
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How to solve an ODE where the rate is directly proportional to two amounts?
Two chemicals in solution react together to form a compound: one unit of compound is formed from $a$ units of chemical $A$ and $b$ units of chemical $B$, with $a + b = 1$. Assume the concentration ...
- 4,450
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2
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Query regarding approach to solve a given differential equation.
There's a equation
$$N(t) = N(t)\frac{P(t,z)}{B}-C\frac{d(P(t,z))}{dz}$$
$$N(t) = A\frac{dP(t,z)}{dt}$$
Constants:
B,
C=3.9878*10⁻⁷,
$A=0.11941$,
Variables:
N(t) is a function of t and is defined at a ...
- 101
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0
answers
21
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Calculating Rate of Change and using differentials to project 3 years from now
Currently, BC is helping $R=5,000$ refugees. The number of refugees that BC must help is rising at a rate of $\frac{dR}{dt}=1,000$ refugees per year. Currently, the number of staff members is $N=100$ ...
- 785
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80
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Simulating Particle motion on a surface
I am working on a personal project to model the motion of a particle on a surface.
Using calculus, I parametrized a surface and then found the normal vector to that surface.
Using that normal vector, ...
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2
votes
1
answer
933
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Is state space representation useful for nonlinear control systems?
I understand that the state space representation is mathematically equivalent to the transfer function representation for linear systems, and that it allows us to solve the corresponding DE by finding ...
0
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1
answer
91
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Understanding and Applying the Half Life Formula
Struggling with this question here:
"One percent of a substance disintegrates in $100$ years. What is its half
life?"
I'm not understanding how to apply the formula $T=\dfrac {\ln 2}k$ to ...
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0
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58
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Simplest application of Picard-Lindelöf in the sciences
I am teaching single-variable real analysis and I want to give the students a concrete example of application of the Picard--Lindelöf theorem for a first-order ODE
$$
\frac{dx}{dt}=f(t,x),$$
where $t$ ...
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0
answers
41
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Sequence of Logic in Diffusion Problem DQ
Problem: If a tank is filled with 100 gallons of water and mistakenly added 300 pounds of salt. To fix the mistake the brine is drained at 3 gallons per minute and replaced with water at the same rate....
2
votes
1
answer
203
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Does there exists two differentiable functions $f, g$ on $I$ such that $W(f, g) (x) >0$ on $A$ and $W(f, g) <0$ on $I\setminus A$?
Let $I=(0, 1) $ and $A=\mathcal{C}\cap (0, 1) $ where $\mathcal{C}$ denote Cantor set.
$\color{red}{Question}$ : Does there exists two differentiable functions $f, g$ on $I$ such that $W(f, g) (x) >...
- 13k
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Does there exist two functions $f, g\in C^1(I)$ for which $W(f, g) (x) >0$ for some $x$ and $W(f, g) (x) <0$ for some $x$?
$f, g\in C^1(I) $ where $I$ is an open interval and $f, g$ both are real valued.
Let $W(f,g)(x) =\begin{vmatrix}f(x) &g(x) \\f'(x)&g'(x)\end{vmatrix}$ denote the Wronskian of $f, g$ at $x\in I$...
- 13k
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55
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Using an expression and an equation to get an ODE to describe something.
I have an expression and an equation, that I need to use to show that ODE describes something.
Let me put it into context
I have an expression for the Rate at Anti-Freeze flows $\mathcal{IN}$
and $\...
1
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1
answer
312
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Using integration to find the population $x$ after a time $t$ years. Having a problem with getting a negative log input.
I'm a little bit confused by a question I came across. It says:
If there were no emigration the population $x$ of a county would increase at a rate of $2.5 \%$ per annum.
By emigration a county loses ...
- 71
1
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1
answer
294
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Question in population dynamics using exponential growth rate equation
Given population doubles in 20 minutes, what is intrinsic growth rate r?
Attempt: Given population doubles, using exponential growth rate we have $\frac{dN}{dt}=2N$ so $N(t)=N_0e^{2t}$ therefore r=2, ...
- 395
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31
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Determining correlations of derivatives of a function given only measurements of that function
Cross-posted from statistics stackexchange:
Say we have a permanent-magnet DC motor that roughly obeys the system equation $\ddot{x}(t) = \alpha \dot{x}(t) + \beta u(t) + \gamma$, where $x(t)$ is the ...
- 1,378
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1
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112
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How to solve simple differential equation (biology)
First of all, I am a biologist and I am not really knowledgeable in mathematics. Thus, I apologize if what I am asking is naive or not fully explained.
I am trying to solve analytically a differential ...
2
votes
0
answers
239
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Thomas Calculus wrong question on Differential Equation?
The problem:
An antibiotic is administered intravenously into the bloodstream at a constant rate $r$. As the drug flows through the patient's system and acts on the infection that is present, it is ...
- 1,156
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1
answer
121
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Ray tracing in nonuniform media; did I write this second order differential equation as two first order differential equations correctly?
Both answers to the Physics SE question Ray tracing in a inhomogeneous media* arrive at some form of the equation below and one links to Florian Bociort's dissertation Imaging properties of gradient-...
- 1,893
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219
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Could there be exact solutions to the Lane-Emden equation for real n≥0 other than 0, 1, or 5?
This Astronomy SE answer says
With a constant $k$ and the polytrop index $n$. This is a result of the solutions of the Lane-Emden equation
$$\frac{1}{\xi^2} \frac{\mathrm{d}}{\mathrm{d}\xi} \left(\xi^...
- 1,893
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1
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397
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Modelling interest with differential equations (IVP)
Problem : you set a bank account, with initial value k, the bank will pay you continuous interest of 12% per year.
a) write an initial value problem for your account balance y(t) after t years
Sol:
$$...
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2
answers
878
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1st order linear differential equation application in electric circuits
I have the following 1st order linear differential equation: $$L\frac{dI}{dt}+RI=E_0\sin(wt).$$
where $L$, $R$ and $E_0$ are constants. The goal here is to discuss the case when $t$ increases ...
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1
answer
264
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Mathematical expression for physical forces in pendulum ODE
A 16 lb weight is suspended from a spring having a spring constant of 5 lb/ft. Assume that an external force given by
24 sin (10t) and a damping force with damping constant 4, are acting on the spring....
- 612
2
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1
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54
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Hyperbolastic rate equation of type II already has its initial condition in it?
I'm modelling some real-world gene expression data with various growth models including linear, exponential, and Verhulst growth but not all of the genes are showing these forms of time-dependence. ...
- 1,876
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427
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Finding the formula for T from Newton's Law of Cooling
I think I got a wrong answer because I skipped a particular step which seemed optional. I'm still not too sure what happened though and would appreciate your help...
Background:
Newton’s law of ...
- 13
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1
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50
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Analytic method for ODE problem
I am studying on a drag force ODE. My question is:
Is there any analytic method to solve $$\frac{dv}{dt}+\alpha v^n=g\\ n \in(1,2]$$ It is somehow look like Bernoulli Differential Equations $y' + p\...
- 25.2k
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121
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Do repeated roots (and Real Jordan form) for ODE's come up in real world applications of ODE's
An equation like $y^{\prime \prime} + 2 y^{\prime} + y = 0$ has repeated roots: The characteristic polynomial is $r^2 + 2r + 1$ which has repeated roots $(-1,-1)$. Two basic solutions of the ODE are ...
- 705
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37
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Help with understanding a differential equation model related to food supply per capita
The growth rate of a population can depend on many factors. For
example, it can depend on the amount of food per capita $A$. If $A_0$
is the mimimum amount of food required, one can think of the ...
- 1,598
1
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1
answer
55
views
Name and application of a nonlinear ODE
Is there a name for an ODE taking form:
\begin{equation}
\left(\frac{dy}{dx}\right)^2 + a y = 0,
\end{equation}
and if there is, what is the constant `a' called either generally or in certain ...
- 502
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1
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44
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The differential equation of the change of the amount of the nutrition inside of the incubator. [closed]
The problem statement is as below.
We will handle the incubator with the microbes inside of it.
The microbe increases consuming a nutrition.
The amount of the nutrition is never increased.
$10^3*P(t):=...
user802763
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votes
1
answer
36
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Finding out the transer function where the equation of the input function and the output function are given
$G(s):=$The transfer function which I want to find out.
$f(t):=$The input function of some system.
$x(t):=$The output function of some system.
The below equation being given.
$4\frac{dx(t)}{dt}+3x(t)=...
user802763
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vote
0
answers
45
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Conceptual complex dynamics - Is it reasonable to perform bifurcation diagrams on PDE's?
I am working on a PDE model, the subject has been modeled with ODE's before. The articles usually have a bifurcation diagram, and to be able to validate my model, I want to compare my work with the ...
- 53
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1
answer
210
views
Criterion to see if you can neglect air drag in projectile motion
In physics education you often consider "real world problems" with projectile motion. Most times in introductory courses you neglect air drag. But how can students (knowing nothing about ...
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92
views
Derive the equations of motion and determine whether angular momentum is conserved..
Suppose that the gravitational force is not given by the inverse-square law, and instead is
$$ F_{grav}=\left(\frac{A}{r^{2}}+\frac{B}{r^{4}}\right)\hat{r}, $$
where A and B are real constants. Derive ...
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0
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46
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Differential equation undamped system
Question image
How can I solve this question?
An object weighting 8 pounds is suspended from a spring stretching it 0.5 feet. The weight, which
is at rest in the equilibrium position, is struck an ...
- 11
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64
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Differential Equation Application of Physics-Sliding Block
Where I can I start to solve this question?
A block is released with an initial velocity v0 = 20 m/s from the bottom of an inclined plane
making an angle of 30◦ with the horizontal. If the constant ...
- 11
1
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1
answer
145
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Basic ODE story - tank with pumps
Full tank has $500$ liter of water containing $0.2\%$ of salt. One pipe pumps clear water in ($100$ liters per minute) and the other gets the mixture from the tank ($100$ liters per minute too). ...
- 2,190
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103
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Calculating if Romeo and Juliet will stay together always or not. [closed]
I have two equations which describe the Love of Romeo for Juliet (R) and Love of Juliet for Romeo (J) as a function of time, $t$.
$R=-c_1e^{3t}-c_2e^{2t}$
$J=2c_1e^{3t}+c_2e^{2t}$
They will stay ...
- 25
1
vote
1
answer
39
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perfect lens: expression for interface
Suppose I want to find a function $(x,z) \mapsto y((x^2+z^2)^{1/2})$ which must represent the interface between two translucent media with different refractive index and which must yield a perfect ...
- 2,455
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votes
2
answers
3k
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Applications of the Laplace Transform
I am curious to know what kind of applications the Laplace transform has. Yes, I know people will reference Wikipedia, and other online sites which discuss the Laplace transform at length. However, ...
- 5,190
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87
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Approximation for Lambert W containing exponential, to help simplify ODE
I have a linear ODE derived from electrical engineering of the form:
$$\omega\cos(\omega t) - W\left(\frac{e^{\omega\cos(\omega t)}}{C}\right) = \frac{Ai'(t)}{i(t)}$$
Where A, C are constants, and W ...
- 2,323
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1
answer
46
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Which logistic equation is better for solving this question?
So I was given a question about spread of disease:
A virus is spreading through a city of 50,000 people who take no precautions. The virus was brought to the town by 100 people and it was found ...
- 13
1
vote
1
answer
444
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Applications of coupled systems of $\;2\times 2\;$ linear differential equations
I am providing maths help to some students studying just before University level in mathematics.
I am writing some practice questions for them on solving coupled first order linear equations and I ...
- 3,182
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1
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211
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Solutions of the Falkner Skan Equation for Negative Beta
I was wondering what the solution to the Falkner Skan equation, $$f^{\prime\prime\prime}+ff^{\prime\prime}+\beta(1-f^{\prime 2})=0$$ looks like for negative values of $\beta$ and whether solutions for ...
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1
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54
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Interpreting solutions to spring-mass ODEs.
I have the following spring-mass ODE solutions:
$$1:\;\;\;x(t)=-3\sin(2t)+4\cos(2t)+12t\sin(t)$$
$$2:\;\;\;x(t)=6e^{-t}\cos(3t)-3e^{-t}\sin(2t)+40\sin(7t)$$
How is is possible to figure if each one is ...
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answer
640
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What are some practical applications of successive differentiation?
Before starting to learn something, I always wonder whats its application. So would you please give some practical examples of application of Successive differentiation and concepts related to it such ...
- 185
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1
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99
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Provide a full phase plane analysis for the model
Provide a full phase plane analysis for the model:
$\left\{\begin{array}{l} \epsilon\dfrac{dx}{dt}=-(x^3-Tx+b)\;,\;T>0\\\dfrac{db}{dt}=x-x_0\end{array} \right.$
So I'm trying to find critical ...
- 462
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vote
2
answers
5k
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Solution to ODE from Newton's Second Law
I have attempted to explore Newton's second law (F = ma) further into its many differential forms. I am not very familiar with differential equations and was searching for the steps and methods to ...
1
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0
answers
36
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Applications of differential equations that boil down to tridiagonal matrix when solved using implicit methods
I am looking for actual applications where differential equations are solved using implicit methods that boil down to solving a tridiagonal every time step. I found that there are heat equations and ...
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