All Questions
26
questions
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 $$\int\frac{e^{2/x}}{(x^2+2x+2)...
- 411
0
votes
0
answers
96
views
Simplifying an arbitrary constant.
Could someone explain me this simplification?
I cannot understand exact reason why $c_1$ is before $\exp$ function without being in another one.
Screenshot presents end of solution of this ...
0
votes
1
answer
229
views
Solving $\frac{dN}{dt} = rN$ (population growth equation) question about using integration to solve simple separable differential equation
I covered differential equations a long time ago and so was confused when my Population Ecology textbook displayed the following "proof" to solve $\frac{dN}{dt} = rN$ for $N$.
My (probably ...
- 887
0
votes
0
answers
46
views
Don't understand how a power of e ends up as a factor in from of e
I have a very basic issue that I can't seem to wrap my head around: I am trying to solve the membrane equation
$$\tau \frac{dV(t)}{dt} = -V(t) + E(t)$$
With a time constant $\tau$, the voltage $V(t)$ ...
- 1
1
vote
1
answer
103
views
Approximating integrals with a sharply peaked integrand
I am working through a textbook on laser trapping and cooling (by Metcalf and van der Straten), but I have purely mathematical question. During a derivation they arrive at the following differential ...
- 189
0
votes
1
answer
127
views
Integrating the product of an exponential and a derivative
I have the following problem that I'm unsure how to tackle:
$\frac{dm}{dt} = \frac{dn}{dt} - \lambda m$
I tried using the integrating factor method with IF = $e^{\lambda t}$ so I end up with:
$me^{\...
- 5
1
vote
2
answers
63
views
Cauchy problem: $x'=\frac{x}{t^2+1}, x(0)=1$.
$$x'=\frac{x}{1+t^2}, \qquad x(0)=1$$
I know the solution to the problem, but I don't get to the right solution myself. My solution:
$$\int \frac{dx}{x}=\int \frac{dt}{1+t^2}$$
$$\ln(x)=\cot(t)+ C$...
- 1,412
1
vote
2
answers
4k
views
Differential equation for bacterial growth
I am assigned with a question which states the rate of a microbial growth is exponential at a rate of (15/100) per hour. where y(0)=500, how many will there be in 15 hours?
I know this question is ...
- 111
0
votes
0
answers
164
views
Derivation of Padé approximant to exponential function: unclear step in Gautschi
On pages 363 to 365 in “Numerical Analysis” by Gautschi (2012) is a derivation of the Padé approximant of the exponential function.
I am stuck on a step in the beginning of the differentiation below (...
- 231
0
votes
2
answers
42
views
Problem with an integral and integration by part
I have a little problem with this question.. :
We have :
$$ G(t) = \int_0^t f(e^t)e^{-t} \, dt $$
With f continue in the domain [0,$+\infty$].
and the question is :
Prove that
$$\int_0^t G(u) \,...
2
votes
2
answers
113
views
General solution of $\frac{dy}{dx}+y=1$, $(y\neq 1)$
Solve $\frac{dy}{dx}+y=1$, $(y\neq 1)$
The general solution for this differential equation is given in my reference as $y=1+Ae^{-x}$, but is it a complete solution ?
My Attempt
$$
\frac{dy}{dx}=1-y\...
- 7,674
-2
votes
1
answer
72
views
Find value for $x$ for which $e^y \frac {dy}{dx} = 6$ [closed]
Find value for $x$ for which
$$e^y \frac {dy}{dx} = 6$$
All help appreciated. All working please.
Thanks.
0
votes
0
answers
75
views
Solving an lineare integro Differential equation
My integro differential equation reads:
$\dot{f}(t)=-\int_0^t\mathrm{d}s\text{ }g(t-s)f(s)$
with $g(t)=\frac{\text{exp}(i\gamma t)}{(1+it)^2}$
with $t\ge 0$, $\gamma>0$.
I tried to solve it with ...
- 191
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
1
vote
2
answers
1k
views
Solving First Order ODE using the integrating factor approach
I am trying to solve the differential equation, but I do not understand the method. Here is my working:
- 305