All Questions
5
questions
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
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
$$ \int \frac{1}{z(1-z)...
- 565
0
votes
1
answer
701
views
First order non-linear ODE with error function
I have to solve $ y'(x)=-2xy(x)+ey^2(x) $.
Using $ z=y^{-1}$ and $-z^{'}=\frac{y^{'}}{y^{2}}$ i arrive to prove that $ z^{'}=-2xz+e $, but when i apply the variation of constants method i obtain $ ...
- 565
3
votes
2
answers
139
views
What type of functional equation is this?
I'm trying to solve the following functional equation
$f\left(x\right)=A\mbox{ exp}\left\{ \int\frac{1}{f\left(x\right)x^{2}+Bx}dx\right\}$
where $f\left(x\right):\mathbb{R}_{+}\rightarrow\mathbb{R}...
- 33
6
votes
1
answer
411
views
Solution of the IVP: $\,y'=\mathrm{e}^{-y^2}-1,\, y(0)=0$
Consider the initial value problem
$$
\frac{dy}{dx} = \mathrm{e}^{-y^2} - 1,\quad y(0)=0.
$$
The Method of Separation of Variables provides that:
$$
\int \frac{dy}{e^{-y^2} - 1} = x+c.
$$
I would ...
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