All Questions
Tagged with ordinary-differential-equations exponential-function
235
questions
3
votes
3
answers
390
views
How can I derive $~\frac{d}{dx}\left(\exp\left(\int f\left(x\right)dx\right)\right)=\exp\left(\int f\left(x\right)dx\right)\cdot f\left(x\right)~$?
$$ P:=\text{function which only contains } ~x~ \text{as variable} $$
$$ I:= \exp\left(\int P dx\right) $$
I want to derive the below equation .
$$ \frac{ d }{ dx } \left( \exp\left(\int P dx\...
- 1,519
2
votes
0
answers
38
views
Exponential growth in contagious disease models
I am a student at the early stage of learning differential equations. In my textbook there is an introduction of the SIR model, and later I also found this Covid prediction model published in 2020.
...
- 21
1
vote
0
answers
52
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Two dimensional oscillation movement: For what conditions does the solution of these Differential Equations give such trajectories?
If the movement equation is like this
two dimensional movement equations:
where $x,z,t$ are variables; the other expressions are constant parameters.
For what conditions is the solution of this ODEs ...
- 11
0
votes
0
answers
19
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Basic Ordinary Differential Equation
Is this first linear ODE? I'm quite confused because the y is in the position of exponential of e.
- 1
0
votes
1
answer
83
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differential equations, exponential population growth
If p is population and t is time. Does that mean that when you do dp/dt = 0 you can find the maximum and minimum population
- 21
0
votes
0
answers
69
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differential equation of a population growth and change - another question
I formulate a system of equations and initial conditions of the following data:
Each year the population1 grows by 4% and population2 by 2%.
Also each year 3% of population1 leaves it and go to ...
- 1,298
0
votes
2
answers
266
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differential equation of a population growth and change
I want to formulate a system of equations and initial conditions of the following data:
Each year the population1 grows by 4% and population2 by 2%.
Also each year 3% of population1 leaves it and go ...
- 1,298
2
votes
1
answer
106
views
Computing $\exp(tT)$ where $T(b)=\{\frac{1}{2m}p^2,b\}$
I've recently been reading Baez's paper on Noether's theorem and I was trying to test the simplest case I could think of: a mass moving in one dimension in a space with potential $0$. The Hamiltonian ...
- 156k
3
votes
1
answer
99
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Name of "divided difference" transform $\frac{f(x)-f(x_0)}{x-x_0}$ and special case $\frac{e^x - 1}{x}$?
Given an analytic function / formal power series
$$\displaystyle f(x)=\sum _{n=0}^{\infty }\frac{f^{(n)}(x_0)}{n!}\left(x-x_{0}\right)^{n}=f(x_0)+f'(x_0)(x-x_{0})+ \tfrac{1}{2}f''(x_0)(x-x_{0})^{2}+\...
- 11.8k
0
votes
0
answers
71
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Estimating parameters of SIR model and problem with real-life data
I tried to make an SIR model based on real-world data. But, I ran into a snag when I'm trying to estimate the parameters of $\beta$ and $\gamma$. With equations:
$$
\begin{cases}
\frac{dS(t)}...
0
votes
2
answers
298
views
Is $y(x)=0$ a solution to the differential equation, $y=y'$?
I think I read or was told that the natural exponential function, $e^x$ is the only solution to $y=y'$, and that it originally was defined by that property.
But isn't $y(x)=0$ one too?
If so, $e^x$ ...
1
vote
1
answer
72
views
Proving matrix exponential
Can anyone tell me how the following is derived?
where $A$ is a matrix.
0
votes
0
answers
47
views
can you help if I'm right on how I found the life time of the fossil?
the question says The half-life of a certain radioactive isotope is approximately $6100$ years. Suppose a fossil is found today to have $0.01\%$ of it’s original amount of this radioactive isotope. ...
- 107
0
votes
2
answers
75
views
Can you help with this exponential decay question?
Suppose that 100 kg of a radioactive substance decays to 80 kg in 20 years.
a) Find the half-life of the substance (round to the nearest year).
b) Write down a function $y(t)$ ($t$ in years) modeling ...
- 107
-4
votes
2
answers
47
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Inverse $D$ operators question
Find $$\dfrac 1 {D^2+6D+9}e^{-3x}$$
So I am new to the topic of $2$nd order linear ordinary differential equations and on $d$-operators. I attempted the question below and am I not supposed to just ...
