All Questions
Tagged with ordinary-differential-equations exponential-function
235
questions
3
votes
2
answers
69
views
$\frac{\text {d}y}{\text {d}x} = e^y$ general solution $y = -\ln(-x+C)$ or $y = -\ln|-x+C|$?
Is the general solution for $\frac{\text {d}y}{\text {d}x} = e^y$
$$y = -\ln(-x+C)$$
or
$$y = -\ln|-x+C|$$
or something else?
Here are the steps I'm taking:
$$\begin{align} \frac{\text {d}y}{\text {d}...
- 133
5
votes
3
answers
1k
views
Solve the differential equation that define exp(x)
In the wikipedia page for the exponential function in the "formal definition" section I found this statement:
Solving the ordinary differential equation $y'(x)=y(x)$ with the
[initial ...
- 277
11
votes
2
answers
680
views
Implicit function equation $f(x) + \log(f(x)) = x$
Is there a function $f \colon \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ such that
$$
f(x) + \log(f(x)) = x
$$
for all $x \in \mathbb{R}_{>0}$?
I have tried rewriting it as a differential equation ...
- 2,005
1
vote
0
answers
31
views
$f(x) = \sum_{j=1}^{4}c_{j}e^{r_{j}x} \quad \text{and} \quad g(x) = \sum_{j=1}^{4}d_{j}e^{s_{j}x}$. How to determine $c_{j},d_{j}$?
Let $\alpha_{i},\beta_{i} \in \mathbb{C}$, $\forall i= 1,2$ and $0 < a < b < \infty$.
\begin{align}
&f^{\prime \prime} + \alpha_{1} f = \alpha_{2} g^{\prime} \quad \text{in} \quad [a,b] \\...
- 657
0
votes
3
answers
112
views
Is there a solution for this non-linear ODE involving exponentials?
There is an non-linear ODE that pops out of the equations a lot when trying to solve the linear case of some second order ODE's.
The equation is this:
$$\ddot{y}+\dot{y}^2=y^2$$
It's easy to see that, ...
5
votes
1
answer
115
views
Proving my IVP for a Piecewise Decay Function (Diff Eq)
Setup
So... I kinda handled most of my proof but I need help with some of the stuff I just kinda went with until it worked out. The problem relates to medicine and its decay in the body. We are given ...
- 51
2
votes
1
answer
220
views
Find $f(x)$ : $ f'(x) = f(x)^2 + f^{-1}(x) + \int_{x}^{-\infty} \frac{e^t}{t} \, dt $
\begin{align}
f'(x) &= f(x)^2 + f^{-1}(x) + \int_{x}^{-\infty} \frac{e^t}{t} \, dt
\end{align}
How to find $f(x)$
What i do so far
\begin{align} f'(x) &= f(x)^2 + f^{-1}(x) + \int_{x}^{-\infty}...
0
votes
0
answers
45
views
Trouble getting the same answer as the textbook (separable first order differential equation)
I was trying to resolve the following differential equation:
$$ y'=e^x(y+1)^2$$
where $y = y(x)$
and I start to resolve it using the following steps:
first we find the solution for when $(y+1)^2=0$ ...
2
votes
1
answer
86
views
Second Order ODE and integral of exponential divided by a polynomial
My original question was
Solve $$x^2y'' + 2y' - 2y = 0$$
First I noticed that $x^2+2x+2$ is a solution. Using order reduction, doing $y = v(x)(x^2+2x+2)$, I found that $$\int\frac{e^{2/x}}{(x^2+2x+2)...
- 411
1
vote
1
answer
56
views
How to calculate the convolution product of $(H_0 e^{\alpha t}) * (H_0 e^{-\alpha t})$?
everyone!
I am doing an exercise concerning the resolution of differential eaqtion in the sense of distributions.
Let $\alpha \in \mathbb{R}_+^*$. Let be the differential equation, for $f \in \mathrm{...
- 25
0
votes
1
answer
66
views
How to prove exponential functional identity knowing that it is a solution to a first order ODE and knowing its Taylor expansion
Establish the identity
$$E(ax)E(bx) = E[(a+b)x]$$
knowing that
$y = E(px)$ satisfies $y' - py = 0$ and
$E(px) = \sum_{n=0}^\infty\frac{(px)^n}{n!}$
An additional hint the textbook gives ; "...
- 1
0
votes
0
answers
96
views
Simplifying an arbitrary constant.
Could someone explain me this simplification?
I cannot understand exact reason why $c_1$ is before $\exp$ function without being in another one.
Screenshot presents end of solution of this ...
2
votes
3
answers
482
views
Guess the particular solution to an exponential function`?
Solve this differential equation $y''+2y'+y = e^{-t}$.
I got the homogenous solution to be
$y_h= (Bt + C)e^{-t}$
But I don't know what my guess to the particular function should be?
$Ae^{-t}$ ...
- 1,237
0
votes
1
answer
24
views
Finding suitable $x:[-T,T]\to\mathbb{R}^n, A\in\mathbb{R}^{n\times n}$ such that $x'(t) = Ax(t)$ when $\frac{d^2}{dt^2}B(t)=B(t)$
Suppose that $\frac{d^2}{dt^2}B(t) = B(t)$ for some matrix $B$ when $t\in [-T, T], T > 0$. I am tasked to determine suitable $x:[-T,T]\to\mathbb{R}^n, A\in\mathbb{R}^{n\times n}$ such that $x'(t) = ...
- 1,221
0
votes
3
answers
187
views
Solutions for $f'=\lambda f$
I am trying to figure out the following problem: Show that $f'=\lambda f$ for a real constant $\lambda$ has only $ce^{\lambda x}$ solutions.
My work: We take a look at $g(x)=f(x)\exp(\lambda x)$. We ...
- 81
1
vote
0
answers
45
views
Tan of sum using differential equations
Define $S(x)$ and $C(x)$ as exponential functions, and $T(x) = S(x)/C(x)$. We want to derive the formula for $T(x+y)$, but the exercise I'm working on suggests setting up a differential equation $g(T(...
- 133
3
votes
1
answer
309
views
Recommendations about the exponential function
I am studying differential equations and I am very surprised by how omnipresent the exponential function is. It pops up everywhere, but there isn't usually a lot of detail provided in introductory ...
- 167
0
votes
1
answer
229
views
Solving $\frac{dN}{dt} = rN$ (population growth equation) question about using integration to solve simple separable differential equation
I covered differential equations a long time ago and so was confused when my Population Ecology textbook displayed the following "proof" to solve $\frac{dN}{dt} = rN$ for $N$.
My (probably ...
- 887
0
votes
1
answer
45
views
The exponential term in a differential equation
Given the differential equation
$P = \frac{L}{De^{-mt} + 1}$,
an intial population of 1000, $L = 10000$ and $m = 0.003$, find the number of years it will take for the population to triple.
To solve ...
- 41
1
vote
0
answers
71
views
Family of straight lines tangent to $e^x$
I just started to learn about Differential equations, I've been reading about families of curves, but I I just got stuck with this problem. I have been Trying to solve it, I have come to find the line ...
- 11
0
votes
0
answers
46
views
Don't understand how a power of e ends up as a factor in from of e
I have a very basic issue that I can't seem to wrap my head around: I am trying to solve the membrane equation
$$\tau \frac{dV(t)}{dt} = -V(t) + E(t)$$
With a time constant $\tau$, the voltage $V(t)$ ...
- 1
5
votes
2
answers
429
views
Exponential growth of a cow farm with constraints in Minecraft
This question is distinct from Exponential growth of cow populations in Minecraft in that an important constraint present in Minecraft is missing from that post. Here are the following constraints:
...
- 861
0
votes
2
answers
80
views
How can I handle$~\exp(\ln|x|)~$to solve 1st order linear DE?
RHS and LHS are same.
$$\exp\left(\ln\left(x\right)\right)=\exp\left(\ln\left(x\right)\right)\tag{1}$$
Taking log.
$$\ln\left(\exp\left(\ln\left(x\right)\right)\right)=\ln\left(\exp\left(\ln\left(x\...
- 1,519
2
votes
0
answers
48
views
if $\dfrac{d\phi}{dx} \leq k \phi$, $\phi(0) = 0$ then $\phi(x) \equiv 0$.
I need a hint (not a full solution, please) on how to prove the following result:
Prove that if $\left\{ \begin{array}{l} \dfrac{d \phi}{dx} \leq k \cdot \phi\\
\phi(0)=0\end{array}\right., k>0$, ...
- 303
0
votes
2
answers
71
views
Alternative argument to show that function diverges everywhere
Consider the function:
$$f(x) = x^{\frac{1}{2}} + \frac{1}{2}x^{-\frac{1}{2}} +\frac{2}{3}x^{\frac{3}{2}} - \frac{1}{4}x^{-\frac{3}{2}} +\frac{1}{15}x^{\frac{5}{2}} + \cdots$$
which is constructed ...
- 2,466
0
votes
1
answer
71
views
Intuitively, why does growth proportional to the population size not diverge but growth proportional to pairs diverges to infinity in finite time?
It's interesting that growth which is proportional to the current population doesn't diverge to infinity, while growth that is proportional to higher powers of the current population.
That is, the ...
- 1,489
0
votes
0
answers
47
views
Simplification of series (A good approximation will also work)
I have solved a system of linear ODEs and obtained a solution of the form
$
y_{ij}(t)=\sqrt{\frac{\gamma^{i(i-1)}}{\gamma^{j(j-1)}}}(\frac{-2\epsilon}{\alpha})^{i-j} \sum_{h=j}^i\frac{e^{-\frac{\alpha ...
- 61
0
votes
1
answer
202
views
Radio active decay formula is not growth formula, $p = p_0 * e^{kt}$, what is the formula for radio active decay, and why?
Trying to solve this question:
A radioactive material is known to decay at a yearly rate proportional to the amount at each moment. There were 2000 grams of the material 10 years ago. There are 1990 ...
- 671
2
votes
1
answer
58
views
Exponential populations that depend upon each other
I have a question about how to solve an exponential problem that involves two populations, each which depends on the other.
For example, let's say we have an initial population of $h$ humans that ...
- 153
0
votes
1
answer
62
views
Why does this data not line up with the differential equation that's supposed to model it?
Sorry for the bad title, I wasn't sure how to ask this specific question.
So for a (extra credit) homework assignment, I wrote a python program for my differential equations class that should model ...
3
votes
3
answers
390
views
How can I derive $~\frac{d}{dx}\left(\exp\left(\int f\left(x\right)dx\right)\right)=\exp\left(\int f\left(x\right)dx\right)\cdot f\left(x\right)~$?
$$ 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
views
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
views
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
views
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
views
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
views
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
views
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
$$\displaystyle f(x)=\sum _{n=0}^{\infty }\frac{f^{(n)}(x_0)}{n!}\left(x-x_{0}\right)^{n}=f(x_0)+f'(x_0)(x-x_{0})+ \tfrac{1}{2}f''(x_0)(x-x_{0})^{2}+\...
- 11.8k
0
votes
0
answers
71
views
Estimating parameters of SIR model and problem with real-life data
I tried to make an SIR model based on real-world data. But, I ran into a snag when I'm trying to estimate the parameters of $\beta$ and $\gamma$. With equations:
$$
\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
views
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 ...
0
votes
1
answer
54
views
An explicit equation for a special damped sinusoid.
I apologize if my question seems weird.
The below damped sinusoid can be described by the following equation:
$$y(t) = A e^{\lambda t} \sin(\omega t)$$
Is it possible to manipulate this equation to ...
- 585
2
votes
1
answer
65
views
Help to solve $y'=y$, building exp function
I come to ask for help building the exponential function as the solution to $y'=y$.
This question is different from :
Prove that $C\exp(x)$ is the only set of functions for which $f(x) = f'(x)$
...
- 153
2
votes
1
answer
107
views
How do we know $\lim_{t\rightarrow \infty }e^{-st + 4t} = 0 ? $
I'm trying to evaluate an integral, and the final step is to evaluate $e^{-st + 4t}$ at infinity minus $e^{-st + 4t}$ at $10$. (The limits of integration were $\infty$ and $10$.)
To evaluate the ...
- 2,331
0
votes
0
answers
39
views
Exponential convergence of interconnected systems
Given an interconnection of dynamic systems $1,2,\cdots, n$ with $x_1(t), x_2(t), \cdots, x_n(t)$ the corresponding states such that
$\dot{x}_{i+1}(t) = -x_{i+1}(t) + f(x_i(t))$, where $f(x_i(t))$ is ...
- 15
1
vote
0
answers
47
views
Rate Equations Appearing in Ecology - Confusion
In ecology and fisheries science it is common to calculate the rates of growth, natural mortality, fishing mortality, immigration, emigration, etc. using 'instantaneous' rates. I understand ...
- 119