All Questions
Tagged with applications ordinary-differential-equations
117
questions
1
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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
views
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
-1
votes
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
answer
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
answers
44
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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
views
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
934
views
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
votes
1
answer
91
views
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
answers
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 $t$ ...
1
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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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1
answer
201
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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
vote
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
vote
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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1
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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
$$\frac{1}{\xi^2} \frac{\mathrm{d}}{\mathrm{d}\xi} \left(\xi^...
- 1,893
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1
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397
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Modelling interest with differential equations (IVP)
Problem : you set a bank account, with initial value k, the bank will pay you continuous interest of 12% per year.
a) write an initial value problem for your account balance y(t) after t years
Sol:
$$...
- 69
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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 ...
0
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1
answer
264
views
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
answer
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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1
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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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0
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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 ...
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