All Questions
8
questions
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
0
votes
2
answers
80
views
How can I handle$~\exp(\ln|x|)~$to solve 1st order linear DE?
RHS and LHS are same.
$$\exp\left(\ln\left(x\right)\right)=\exp\left(\ln\left(x\right)\right)\tag{1}$$
Taking log.
$$\ln\left(\exp\left(\ln\left(x\right)\right)\right)=\ln\left(\exp\left(\ln\left(x\...
- 1,519
0
votes
0
answers
71
views
Estimating parameters of SIR model and problem with real-life data
I tried to make an SIR model based on real-world data. But, I ran into a snag when I'm trying to estimate the parameters of $\beta$ and $\gamma$. With equations:
$$
\begin{cases}
\frac{dS(t)}...
3
votes
4
answers
72
views
Non-linear second order ODE
I have to solve $$ y''(x)+(y'(x))^2=y'(x). $$
Using $ y'(x)=z $, I can write $$\int \frac{1}{z-z^2}dz=\int dx $$
So:
$$\frac{1}{z(1-z)}=\frac{A}{z}+\frac{B}{1-z}$$
leads to
$$ \int \frac{1}{z(1-z)...
- 565
1
vote
1
answer
115
views
Can modulus function present in a particular solution
Particular solution of $2ye^{\tfrac{x}{y}}dx+\Big(y-2xe^{\tfrac{x}{y}}\Big)dy=0$, $x=0$ when $y=1$
Attempt
Put $x=vy$
$$
\frac{dx}{dy}=\frac{x}{y}-\frac{1}{2e^{\tfrac{x}{y}}}\\
\frac{dx}{dy}=v+y\...
- 7,674
2
votes
2
answers
129
views
Solution of $\frac{\mathrm{d}y}{\mathrm{d}x}=y\mathrm{e}^x$ given $x=0$, $y=\mathrm{e}$
$\dfrac{\mathrm{d}y}{\mathrm{d}x}=y\mathrm{e}^x$, $x=0$ and $y=\mathrm{e}$. Find the particular solution.
Attempt 1
$$
\dfrac{\mathrm{d}y}{\mathrm{d}x}=y\mathrm{e}^x\implies\dfrac{\mathrm{d}y}{y}=\...
- 7,674
0
votes
1
answer
85
views
When will the population of a sample double (using dif-eq)?
I have the initial equation $$\frac{dP}{dt}=kp$$ where P is the population, t is time, and k is some positive constant. The rest of what I'm given is that P(0) = A, what is the time for the population ...
- 77
0
votes
1
answer
127
views
Expotential Growth/Decay - Problem Deriving Atmospheric Pressure Formula
I have a problem deriving the following formula:
$$\frac{dP}{dh} = k\left(\frac{P}{T}\right)$$
Using the following 'rule': If $\ \dfrac{dA}{dt} = kA\,$ then $\,A = A_0\left(e^{\,kt}\right)\,$ ...
- 77