All Questions
Tagged with ordinary-differential-equations exponential-function
236
questions
3
votes
2
answers
64
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:
$$\begin{align} \frac{\text {d}y}{\text {d}...
- 133
-2
votes
2
answers
105
views
Proof that there is only k results for $f(x)=f'(x)$ [duplicate]
In the exponential fonction page of wikipedia, it is stated that $exp(x)$ is the only one result of these conditions for a function $f(x)=f'(x)$ and $f(0)=1$
[![enter image description here][1]][1]
I ...
- 307
7
votes
3
answers
1k
views
Solve the differential equation that define exp(x)
In the wikipedia page for the exponential function in the "formal definition" section I found this statement:
Solving the
ordinary differential equation
$y'(x)=y(x)$ with the
initial ...
- 307
11
votes
2
answers
677
views
Implicit function equation $f(x) + \log(f(x)) = x$
Is there a function $f \colon \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ such that
$$
f(x) + \log(f(x)) = x
$$
for all $x \in \mathbb{R}_{>0}$?
I have tried rewriting it as a differential equation ...
- 2,005
1
vote
0
answers
30
views
$f(x) = \sum_{j=1}^{4}c_{j}e^{r_{j}x} \quad \text{and} \quad g(x) = \sum_{j=1}^{4}d_{j}e^{s_{j}x}$. How to determine $c_{j},d_{j}$?
Let $\alpha_{i},\beta_{i} \in \mathbb{C}$, $\forall i= 1,2$ and $0 < a < b < \infty$.
\begin{align}
&f^{\prime \prime} + \alpha_{1} f = \alpha_{2} g^{\prime} \quad \text{in} \quad [a,b] \\...
- 657
0
votes
3
answers
111
views
Is there a solution for this non-linear ODE involving exponentials?
There is an non-linear ODE that pops out of the equations a lot when trying to solve the linear case of some second order ODE's.
The equation is this:
$$\ddot{y}+\dot{y}^2=y^2$$
It's easy to see that, ...
5
votes
1
answer
115
views
Proving my IVP for a Piecewise Decay Function (Diff Eq)
Setup
So... I kinda handled most of my proof but I need help with some of the stuff I just kinda went with until it worked out. The problem relates to medicine and its decay in the body. We are given ...
- 51
2
votes
1
answer
219
views
Find $f(x)$ : $ f'(x) = f(x)^2 + f^{-1}(x) + \int_{x}^{-\infty} \frac{e^t}{t} \, dt $
\begin{align}
f'(x) &= f(x)^2 + f^{-1}(x) + \int_{x}^{-\infty} \frac{e^t}{t} \, dt
\end{align}
How to find $f(x)$
What i do so far
\begin{align} f'(x) &= f(x)^2 + f^{-1}(x) + \int_{x}^{-\infty}...
0
votes
0
answers
45
views
Trouble getting the same answer as the textbook (separable first order differential equation)
I was trying to resolve the following differential equation:
$$ 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
65
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
95
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
473
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) = ...
- 1,211