All Questions
Tagged with ordinary-differential-equations exponential-function
235
questions
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
1
vote
2
answers
1k
views
Is $dX/dt=X(t)$ the correct ODE for $X(t)=e^t$?
For a school project for chemistry I use systems of ODEs to calculate the concentrations of specific chemicals over time. Now I am wondering if
$$ \frac{dX}{dt} =X(t) $$
the same is as
$$ X(t)=e^...
- 117
-2
votes
1
answer
893
views
Why can $e^x$ be defined as the unique function $f(x)$ such that $f(x)=f'(x)$ and $f(0)=1$?
The definition that $e^x$ is the unique function $f(x)$ such that $f(x)=f'(x)$ and $f(0)=1$ has two problems for me:
How is $e^x$ the unique function that satisfies this property? $ke^x$ also has ...
- 20.7k
1
vote
1
answer
103
views
Approximating integrals with a sharply peaked integrand
I am working through a textbook on laser trapping and cooling (by Metcalf and van der Straten), but I have purely mathematical question. During a derivation they arrive at the following differential ...
- 189
0
votes
1
answer
127
views
Integrating the product of an exponential and a derivative
I have the following problem that I'm unsure how to tackle:
$\frac{dm}{dt} = \frac{dn}{dt} - \lambda m$
I tried using the integrating factor method with IF = $e^{\lambda t}$ so I end up with:
$me^{\...
- 5
0
votes
2
answers
78
views
Find a function $f$ such that $\int_0^{P(x)} f(t) dt = 1- e^{2P(x)}$
I'm trying to solve the following homework problem. It states as follows:
"Let $P(x)$ be a polynomial such that $P'(x) \neq 0$ for all values of $x$. Does there exist a continuous function $f$ such ...
- 7,273
0
votes
1
answer
22
views
estimate quantity function q(p) from log estimates
I'm trying to create a demand curve to measure the demand of an asset as a function of its price. In research I've found others who have determined using empirical data that:
$ ln(q) = -0.7ln(p) $ ...
- 478
-2
votes
4
answers
49
views
How to solve $\frac{\mathrm{d}I}{\mathrm{d}t} = (\beta - \gamma)I$? [closed]
I need help with solving this exponential growth equation:
$$\frac{\mathrm{d}I}{\mathrm{d}t} = (\beta - \gamma)I.$$
- 19
0
votes
1
answer
54
views
Solution of Linear Differential Equations
If $[A, [A, B]] = 0 = [B, [A, B]]$, where $[A,B]=AB-BA$, then
$$e^{tA} e^{tB}=e^{t(A+B)+(t^2/2)[A,B]}$$
The book suggests proving that $e^{t(A+B)+(t^2/2)[A,B]}$ is solution of
$$\dot{X}=AX+XB, \...
- 2,797
0
votes
1
answer
358
views
Second order differential equation with complex coefficient
I have some doubts about this kind of second - order differential equation, which is used a lot in physics and for which there are many topics (but in this case the situation is a bit different ...
- 239
1
vote
1
answer
90
views
How to approach the functional equation $\frac{f(x+T)}{f(x)} = g(x)$?
I am trying (for fun!) to study the ongoing COVID-19 pandemic and have the following question. we know that an exponential function satisfies the following functional equation:
\begin{equation}
\frac{...
- 3,386