All Questions
Tagged with ordinary-differential-equations exponential-function
40
questions with no upvoted or accepted answers
3
votes
1
answer
105
views
Solution for $\int \frac{1}{1-we^w}dw$
I am looking for a solution or a method of approximation for :
$$\int \frac{1}{1-we^w}dw$$
that came up while working on an ODE problem.
Got any suggestions?
Note: $w$ is also a one variable ...
- 143
3
votes
0
answers
43
views
Math question - specifying the range of ln
The original problem is this:
$$\lim_{x\to 0}\frac{1}{2x}\ln \frac{1}{n}\sum_1^n {e^{kx}} = 20$$
Find out the value of $n$, which is a natural number.
But I'm having a bit of trouble solving this ...
- 31
3
votes
0
answers
65
views
Show that $x' = Ax$ is an attractor if end only if there is a quadratic form $q$ positive definite such that $Dq(x) . Ax < 0$ for all $x \neq 0$
Show that $x' = Ax$ is an attractor if end only if there is a quadratic form $q$ positive definite such that $$Dq(x) . Ax < 0$$ for all $x \neq 0$
Definition: a linear system $x' = Ax$ called ...
- 844
2
votes
0
answers
48
views
if $\dfrac{d\phi}{dx} \leq k \phi$, $\phi(0) = 0$ then $\phi(x) \equiv 0$.
I need a hint (not a full solution, please) on how to prove the following result:
Prove that if $\left\{ \begin{array}{l} \dfrac{d \phi}{dx} \leq k \cdot \phi\\
\phi(0)=0\end{array}\right., k>0$, ...
- 303
2
votes
0
answers
38
views
Exponential growth in contagious disease models
I am a student at the early stage of learning differential equations. In my textbook there is an introduction of the SIR model, and later I also found this Covid prediction model published in 2020.
...
- 21
2
votes
0
answers
73
views
differential operator
I've read journal "On the Comparison of Several Mean Values: An Alternative approach" (Welch, 1951). I don't understand this expression:
$$E\left(\exp\left[ \sum_t ( w_t - \omega_t ) D_t\right]\right),...
1
vote
0
answers
31
views
$f(x) = \sum_{j=1}^{4}c_{j}e^{r_{j}x} \quad \text{and} \quad g(x) = \sum_{j=1}^{4}d_{j}e^{s_{j}x}$. How to determine $c_{j},d_{j}$?
Let $\alpha_{i},\beta_{i} \in \mathbb{C}$, $\forall i= 1,2$ and $0 < a < b < \infty$.
\begin{align}
&f^{\prime \prime} + \alpha_{1} f = \alpha_{2} g^{\prime} \quad \text{in} \quad [a,b] \\...
- 657
1
vote
0
answers
45
views
Tan of sum using differential equations
Define $S(x)$ and $C(x)$ as exponential functions, and $T(x) = S(x)/C(x)$. We want to derive the formula for $T(x+y)$, but the exercise I'm working on suggests setting up a differential equation $g(T(...
- 133
1
vote
0
answers
71
views
Family of straight lines tangent to $e^x$
I just started to learn about Differential equations, I've been reading about families of curves, but I I just got stuck with this problem. I have been Trying to solve it, I have come to find the line ...
- 11
1
vote
0
answers
52
views
Two dimensional oscillation movement: For what conditions does the solution of these Differential Equations give such trajectories?
If the movement equation is like this
two dimensional movement equations:
where $x,z,t$ are variables; the other expressions are constant parameters.
For what conditions is the solution of this ODEs ...
- 11
1
vote
0
answers
47
views
Rate Equations Appearing in Ecology - Confusion
In ecology and fisheries science it is common to calculate the rates of growth, natural mortality, fishing mortality, immigration, emigration, etc. using 'instantaneous' rates. I understand ...
- 119
1
vote
1
answer
250
views
Average life expectancy..exponential function
Let
$$N_0 = \text{initial number of AIDS patients}$$
$$N= \text{number of patients left}$$
The equation is given by: $$N=N_0\exp(-kt)$$
What is the average life expectancy of one person?
(The answer ...
- 21
1
vote
1
answer
19
views
Investor estimates a fortune "t" years from now, show initial growth rate
an investor estimates his fortune in "t" years' time will be: (in thousands of dollars)
$ y(t) = 12[(t/10) + 1]^{3/2}$
a) what is the initial growth rate of his fortune? (give as percent per year)
$...
- 21
1
vote
0
answers
134
views
Weird result regarding "infinitely explosive" differential equations
Firstly, take the family of differential equations $\dot x =
> \frac{dx}{dt}=x^\alpha$, for any $\alpha \in \mathbb R$
The solution to these equations is
$$(\text{for } \alpha=1):x(t)=x_0e^t$$
$$(\...
- 13.4k
1
vote
1
answer
926
views
solve initial value problem using exponential matrix
$x'' = 2 x' +6y +3$
$y' = -x' -2y$
subject the the initial condition
$x(0) = 0; x'(0) = 0; y(0) = 1$
The first part of the question is about finding $e^{At}$ of this matrix $A = \begin{bmatrix}
...
