All Questions
Tagged with ordinary-differential-equations exponential-function
22
questions
132
votes
9
answers
27k
views
Prove that $C e^x$ is the only set of functions for which $f(x) = f'(x)$
I was wondering on the following and I probably know the answer already: NO.
Is there another number with similar properties as $e$? So that the derivative of $ e^x$ is the same as the function itself....
- 1,647
12
votes
6
answers
4k
views
Are the any non-trivial functions where $f(x)=f'(x)$ not of the form $Ae^x$
This may seem like a silly question, but I just wanted to check. I know there are proofs that if $f(x)=f'(x)$ then $f(x)=Ae^x$. But can we 'invent' another function that obeys $f(x)=f'(x)$ which is ...
- 785
4
votes
3
answers
1k
views
Prove that if $\phi'(x) = \phi(x)$ and $\phi(0)=0$, then $\phi(x)\equiv 0$. Use this to prove the identity $e^{a+b} = e^a e^b$.
I am given the following. hint Consider $f(x)=e^{-x} \phi(x)$.
I am unsure how to approach this problem.
- 2,649
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
8
votes
4
answers
2k
views
How unique is $e$?
Is the property of a function being its own derivative unique to $e^x$, or are there other functions with this property? My working for $e$ is that for any $y=a^x$, $ln(y)=x\ln a$, so $\frac{dy}{dx}=\...
- 1,750
8
votes
4
answers
7k
views
How to compute time ordered Exponential?
So say you have a matrix dependent on a variable t:
$A(t)$
How do you compute $e^{A(t)}$ ?
It seems Sylvester's formula, my standard method of computing matrix exponentials can't be applied here ...
- 17.5k
7
votes
3
answers
430
views
What are other solutions to this differential equation, "similar" to $\sin x$ and $e^x$?
I've been studying electronics, where they make great use of the relationship between the sine and exponential functions ($e^{i \omega t} = \cos{\omega t} + i \sin \omega t)$. This relationship is ...
- 389
4
votes
2
answers
4k
views
Show that $e^{t(A+B)} = e^{tA}e^{tB}$ for all $t \in \mathbb{R}$ if, and only if $AB = BA$.
Let A,B real or complex matrixes. Show that $e^{t(A+B)} = e^{tA}e^{tB}$ for all $t \in \mathbb{R}$ if, and only if $AB = BA$.
I demonstrated the reciprocal:
$\Leftarrow )$ The two equations are ...
- 1,819
2
votes
2
answers
282
views
Computing Lyapunov Exponents
Consider $\mathcal{T}=S^{1}\times D^{2}$, where $S^{1}=[0,1]\mod 1$ and $D^{2}=\{(x,y)\in\mathbb{R}^{2}|x^{2}+y^{2}\le 1\}$. Fix $\lambda\in(0,\frac{1}{2})$ and define the map $F:\mathcal{T}\to\...
- 1,038
1
vote
1
answer
2k
views
Proof of existence and uniqueness of the exponential function using ODEs
In our lecture notes for our complex analysis class, we were given the following theorem:
Theorem: There exists a unique complex function $f$ such that
$f(z)$ is a single valued function $f(z) \in \...
user100463
3
votes
7
answers
615
views
Intuitive explanation of $y' = y \implies y = Ce^x$
I understand why $f : \mathbb{R} \to \mathbb{R}$ with $f'(x) = f(x)$ and $f(0) = 1$ must be $f (x) = e^x$, but I don't really feel it is super intuitive. Intuitively, why would you expect such a ...
- 1,872
3
votes
1
answer
1k
views
problem about population growth
At the beginning of the Gold Rush, the population of Coyote Gulch,Arizona was $365$.From then on ,the population would have grown by a factor of $e$ each year,except for the high rate of "accidental" ...
- 1,027
3
votes
5
answers
2k
views
Sum of exponential growth and decay
Suppose we have the equation:
$$y(t_i) = C_0 + C_1 e^{-\lambda_1 t_i} + C_2 e^{\lambda_2 t_i}$$
where $C_0$, $C_1$, $\lambda_1$, $C_2$, and $\lambda_2 \ge 0$
This is equivalent to the summation of ...
- 419
2
votes
3
answers
143
views
Explanation of proof nedeed: Why is $y'=c \cdot y$ always a exponential growth/decay function?
Good evening,
I'm struggling with understanding a proof:
I know, that a solution of $y'=c \cdot y$ is $y=a \cdot e^{ct}$ and it's clear how to calculate this.
I want to proof, that all solutions of ...
2
votes
3
answers
10k
views
Newton's law of cooling, soup
Newton's law of cooling states that the temperature $T(t)$ of an object at time $t > 0$ changes at a rate proportional to the difference between the temperature of the object and the temperature $...
- 532