1
vote
1answer
25 views

problem about population growth

At the beginning of the Gold Rush, the population of Coyote Gulch,Arizona was $365$.From then on ,the population would have grown by a factor of $e$ each year,except for the high rate of ...
0
votes
0answers
18 views

Generic Exponential curve base derivation

Alrighty so I am working on a computer program that forms ADSR envelopes including exponential curves for the attack, decay, and release segments. It uses the following equation for the exponential ...
0
votes
1answer
32 views

First order ODE with $f'(x) = 810(10)^x$

I'm trying to find an explicit form of the series $f(0) = 89.1,f(1) = 899.1,f(2) = 8999.1, \cdots$. My first though was to take the derivative and integrate it, which I've done before with a fair ...
2
votes
1answer
46 views

Second order linear ODE not making sense…

I am given: $y''-3y'+2y=0$ $y(0)=1$ $y'(0)=2$ I know that $r_1=2$ and $r_2=1$ The solution therefore is: $y(x)=C_1e^x+C_2e^{2x}$ Solving for initial values, I have: $y(0)=C_1+C_2=1$ ...
0
votes
1answer
53 views

How to solve this system of ODE's?

I'm not sure how to proceed to solve this system of ODE's; $$ \begin{bmatrix}\dot{x}_1 \\\dot{x}_2\end{bmatrix}=\begin{bmatrix} \cos t & -\sin t\\ \sin t & \cos t ...
1
vote
1answer
38 views

Should I use Taylor Expansion or the expansion of $e^x$ to express a second order differential (in this case)

I've been given this equation: $(1+x^2)\dfrac{d^2y}{dx^2} + 4x\dfrac{dy}{dx} + 2y = 0$ I've also been told that: $y=1, \dfrac{dy}{dx} = 1$, at $x=-1$ I've been asked to find a series solution of ...
0
votes
2answers
71 views

A Complex Variable ODE

suppose $f$ is a holomorphic function on some domain $D$ satisfying $f'(z)=af(z) $ for some >constant a. show that $f(z)=Ce^{az}$, for some constant $C$
0
votes
2answers
53 views

Linear system of ODEs

Given is the ODE system $y'=\left(\begin{matrix}1\\1\\0\\ \end{matrix}\right)+\left(\begin{matrix}0&0&0\\0&k&0\\0&-k&k\\ \end{matrix} \right)y$ with boundary conditions ...
2
votes
2answers
80 views

Differentiation of exponential function? [closed]

How to solve derivative $\lim_{n\to\infty}e^{{}^n(x)}$ with respective of $x$ ? Here, ${}^n(x)$ is a tetration function $$ {}^n(x)= \begin{cases} x^{[{}^{n-1}(x)]} & \mbox{ if } {\;n>1}\\ x ...
1
vote
1answer
99 views

How do I solve this exponential decay problem?

This is a problem from page 44 of Edwards & Penneys' Elementary DE Problems, Question #41: Suppose that a mineral body formed in the ancient cataclysm originally containing the uranium ...
2
votes
1answer
90 views

Radioactive Substance Decay Problem

A radioactive substance decays according to : $$x' = -ax$$ where $a>0$ is a constant. After $2$ days there are $1,000$ grams and after $7$ days there are $300$ grams. How many grams were there ...
2
votes
1answer
217 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 ...
3
votes
2answers
72 views

Constructing a differential equation for hyperbolic crochet

There is plenty of information about hyperbolic geometry and its melding with crochet, however I have yet to find an exact equation for determining the number of stitches in each row. I will try to ...
0
votes
4answers
102 views

Finding a differential equation when a half life is known

Does anyone know how I would write a differential equation for the following? I am not interested in the answer as such, I'm more interested in the steps and how to obtain the answer. I don't know how ...
0
votes
1answer
88 views

Commuting Exponential Matrices

Let $x(t)=\exp(tA)\exp(tB)$ and $y(t)=\exp(t(A+B))$. Show that if $AB=BA$ then $x(t)$ and $y(t)$ satisfy the same initial value problem for ODEs and therefore must be equal. $A, B$ square matrices. ...
1
vote
2answers
54 views

Please help to solve this ODE with function coefficients

Is it possible to solve this ODE for $y$? According to wikipedia this falls in the category of a first order, linear, inhomogeneous ODE with function coefficients. But is there a more tractable ...
2
votes
0answers
54 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 ) ...
5
votes
3answers
205 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 ...
1
vote
0answers
164 views

maple code for exp-func. for solving PDE's & non-linear ODE's?

How can I create the Maple code using exponential-function solving the equation below? $u_t = \gamma u_x+6u(u_x)^2+(3u^2-1)u_{xx}-u_{xxxx}$ $u_t =u_{xx}-u^3+u,$ $\alpha u''(x) = \beta ...
1
vote
1answer
217 views

Need to deduce $f(x)$ from $f_x=e^{t(x)}$

I know that $$f_x=e^{t(x)}$$ (where the notation $f_x=\frac{df}{dx}$) (EDIT: $f=f(x)$ and $t$ parameterizes $x$, so $x=x(t) \Leftrightarrow t=t(x)$) and that therefore $$\frac{d^n ...
2
votes
1answer
428 views

Fundamental matrix and exponential of matrix using Laplace Transform

I'm trying to work out how to find $$\exp(At)$$ for a system of linear differential equations $$x'=Ax.$$ I know that the solution is a fundamental matrix of the system such that $$\exp(At)=I$$ at ...