All Questions
Tagged with ordinary-differential-equations exponential-function
235
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
12
votes
1
answer
11k
views
Why is the formal solution to a linear differential equation of exponential form?
So $x(t) = e^{ct}$ solves $dx/dt = cx$. This is clear enough from differentiation rules... But I fail to grasp, in some sense which I can't quite put my finger on, why it is so. Why can the solution ...
- 153
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
10
votes
3
answers
263
views
Physical intuition for the solution to $y' = y$.
Assume that $e^x$ has not been defined, so please do not refer to $e$ in an answer.
Given the D.E $y' =ry$, we substitute a power series and arrive at the solution:
$$y = \sum_{k =0}^\infty \frac{(rx)^...
- 714
9
votes
5
answers
7k
views
What is the derivative of a function of the form $u(x)^{v(x)}$?
So I have a given lets say $(x+1)^{2x}$ in addition to $\frac{\mathrm dy}{\mathrm dx}a^u=a^u\log(a)u'$. I still have to multiply this by the derivative of the inside function $x+1$ correct?
- 307
9
votes
4
answers
259
views
"Natural" proof of $P\left(\frac{d}{dx}\right)\bigl(e^{xy}Q(x)\bigr)=Q\left(\frac{d}{dy}\right)\bigl(e^{xy}P(y)\bigr)$.
In the context of linear differential equations, I've stumbled upon the following identity for an arbitrary pair of polynomials $P$ and $Q$ with real or complex coefficients:
$$
P\left(\frac{d}{dx}\...
- 2,210
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
3
answers
626
views
Solution to differential equations $y(0)=1$ and $y^{(n)}=y+1$
When I was solving some differential equations, I asked myself the following:
Is there a function has the following:
$$y'=y+1$$
$$y''=y+1$$
$$y'''=y+1$$
$$......$$
$$......$$
If the initial value is $...
- 23.4k
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
6
votes
1
answer
411
views
Solution of the IVP: $\,y'=\mathrm{e}^{-y^2}-1,\, y(0)=0$
Consider the initial value problem
$$
\frac{dy}{dx} = \mathrm{e}^{-y^2} - 1,\quad y(0)=0.
$$
The Method of Separation of Variables provides that:
$$
\int \frac{dy}{e^{-y^2} - 1} = x+c.
$$
I would ...
- 2,375
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
5
votes
6
answers
115
views
Need explanation for simple differential equation
I can't figure out this really simple linear equation:
$$x'=x$$
I know that the result should be an exponential function with $t$ in the exponent, but I can't really say why. I tried integrating ...
- 85
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