Let us consider the matrix $A$ which has three parameters $R,C1,C3$. This is from the Ikeda map in real form. It is defined as $$x \rightarrow R+(x \cos(\tau)-y \sin(\tau))$$ $$y \rightarrow x\sin(\tau)+y\cos(\tau)$$ The Jacobain matrix is given by: \begin{equation*} A = \begin{bmatrix} \cos...

The sigmoid function, $S(x) = \frac{1}{1+e^{-x}}$, achieves close to what you need (with appropriate scaling and shifting of the function). Do you need the function to be exactly $\pm 50$ when evaluated at $\pm 10$? If so, a polynomial option would be to use something called the smoothstep func...

Can we think of an Ordinary Differential Equation which has no solution? How can we think of this ODE? Like what $f$ should I use such that $\frac{dy}{dx} = f(x,y)$ with $y(x_{0}) = f(x_{0},y_{0})$. From existence theorem I think I have to use a $f$ which is not Lipschitz continuous?

Let $f:\mathbb{R} \rightarrow \mathbb{R}$. Then consider the following delay equation : $$f'(x) = f(x-1)$$ Let $S$ be the set of solution ot this equation. Then I would like to prove that : $\forall f \in S -\{ x \mapsto 0\}$, $f$ is not bounded. What I've done so far is that : The set of...

While studying Stochastic Process, I came across the following equation which I am trying to reduce it int simple terms! It is $P_{0}(1 + \sum_{n=1}^{(s-1)} \frac{\rho^n}{n!} + \sum_{n=s}^{\infty}\frac{\rho ^n}{s! (s^{n-s})}) = 1$ I have to evaluate $P_{0}$ in simplest form. I tried to see t...

Here is the statement of the problem: Consider the Legendre Equation $$ (*)\qquad (1-x^2)y''-2xy'+2y=0 $$ (a) Find two linearly independent solutions about $x=0$, solving completely any relevant recurrence relations. (b) Compute the radius of convergence for each fundamental solution in part...

Instead of doing that, why don't you just multiply the whole expression by $y$ and the let $y^2=t$.

I was trying to solve this ODE $\frac{dy}{dx} = c_{1} + c_{2}y + \frac{c_{3}}{y} , y(0) = c , c >0$. where $c_{1},c_{2},c_{3}$ are three real numbers say $c_{1} < 0,c_{2},c_{3} > 0$. I thought of using separation of variables giving me $x = \int(\frac{y}{c_{1}y+c_{2}y^2+c_{3}})dy + c$. Next I ...

Well I tried this,may be of some help- $z + z^2 + ... +z^n = n|z|^n$ differentiating the above series wrt $z$ and as right side is a constant for a fixed $n$ so rhs will be $0$ on differentiating $1 + 2z + 3z^2 +...+nz^{n-1} = 0 $ subtracting the latter from the first equation we get- $1 +...