All Questions
33
questions
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 $(y+1)^2=0$ ...
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
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 .
$$ \frac{ d }{ dx } \left( \exp\left(\int P dx\...
- 1,519
0
votes
0
answers
19
views
Basic Ordinary Differential Equation
Is this first linear ODE? I'm quite confused because the y is in the position of exponential of e.
- 1
0
votes
0
answers
69
views
differential equation of a population growth and change - another question
I formulate a system of equations and initial conditions of the following data:
Each year the population1 grows by 4% and population2 by 2%.
Also each year 3% of population1 leaves it and go to ...
- 1,298
0
votes
2
answers
266
views
differential equation of a population growth and change
I want to formulate a system of equations and initial conditions of the following data:
Each year the population1 grows by 4% and population2 by 2%.
Also each year 3% of population1 leaves it and go ...
- 1,298
0
votes
2
answers
298
views
Is $y(x)=0$ a solution to the differential equation, $y=y'$?
I think I read or was told that the natural exponential function, $e^x$ is the only solution to $y=y'$, and that it originally was defined by that property.
But isn't $y(x)=0$ one too?
If so, $e^x$ ...
1
vote
2
answers
1k
views
Is $dX/dt=X(t)$ the correct ODE for $X(t)=e^t$?
For a school project for chemistry I use systems of ODEs to calculate the concentrations of specific chemicals over time. Now I am wondering if
$$ \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
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
0
votes
0
answers
267
views
Alternate proof for the derivative of $e^x$ using L'Hospital's rule – is there a generalization?
Before you say you can't use the derivative of a function to prove the derivative of a function, just please see the proof.
I created it myself, and was wondering whether this could applied to any ...
- 446
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
0
votes
4
answers
64
views
What are differential equations and how do you solve ${dy \over dx}=y$ and find $y$ in terms $x$?
I had been wondering about how to solve the equation $${dy \over dx}=y$$. My progress was to use the chain rule, like setting $$z=2x\;{dy\over dx}={dy\over dz}*{dz\over dx}={dy\over d(2x)}*2=y.$$ Now ...
- 414