Intereting Posts

Using the digits $7,7,7,7,1$ and the operators $+,-,*,/$ to make a formula which equals $100$
Lifting a homomorphism
Multiplication of a random variable with constant
Number of ways of spelling Abracadabra in this grid
Ratio test with limsup vs lim
Grothendieck's “Relative” Point of View
Expressing the integral $\int_{0}^{1}\frac{\mathrm{d}x}{\sqrt{\left(1-x^3\right)\left(1-a^6x^3\right)}}$ in terms of elliptic integrals
Proving that if $T \in \mathcal{L}(V)$ is normal, then the minimal polynomial of $T$ has no repeated roots.
Proof of the independence of the sample mean and sample variance
Banach spaces over fields other than $\mathbb{C}$?
Using Parseval's theorem to solve an integral
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How many essential ways: $n$ balls in $k$ bins, where $k$ can vary and there are $\geq 2$ balls in each bin?
Does the sequence$ f_n(x)=\frac{x}{1+nx^2}$ converge uniformly on $\mathbb{R}$?
Primes for which $x^k\equiv n\pmod p$ is solvable: the fixed version

I came across the following question in one of my professor’s past exams:

Find a Lyapunov function for $(0,0)$ in the system:

$$

\left\{

\begin{array}{ll}

\dot{x} = -x -2y + x^2\\

\dot{y} = x – 4y + xy

\end{array}\right.

$$

I know there is no formula for finding Lyapunov functions for a system, so how do I start solving such problems?

- Exponential of the differential operator
- Can this gravitational field differential equation be solved, or does it not show what I intended?
- To what extent can you manipulate differentials like dy and dt like actual values?
- Optimizing a functional with a differential equation as a constraint
- Book recommendation for ordinary differential equations
- Solutions to $f'=f$ over the rationals

Thanks!

- Blowing-up a singular point
- How to solve differential equations of the form $f'(x) = f(x + a)$
- Omega limit set is invariant
- Finding Lyapunov function for a given system of differential equations
- Can anyone explain the intuitive meaning of 'integrating on both sides of the equation' when solving differential equations?
- Geometric & Intuitive Meaning of $SL(2,R)$, $SU(2)$, etc… & Representation Theory of Special Functions
- Modelling a Forced undamped oscillation via ODE
- Interval of definition of the solutions of $\dot x=e^x\sin x$
- Solving $y' = \sqrt{|y|}$
- Integrating absolute value function

For many ODEs, a good bet is to try a polynomial as a Lyapunov function candidate. In fact, in practice, a common method is to search for a sum-of-squares polynomial, i.e., a polynomial that can be given as $\sum_{i=1}^k p_i(x)^2$, with $p_1, \dots, p_k$ polynomial.

I haven’t tried this example, but a good approach might be to try something of the form $V(x,y) = (x+ay+b)^2 + cy^2 + d$, with parameters $a,b,c,d$.

- Help me put these enormous numbers in order: googol, googol-plex-bang, googol-stack and so on
- Control / Feedback Theory
- Prove that $\cot^2{(\pi/7)} + \cot^2{(2\pi/7)} + \cot^2{(3\pi/7)} = 5$
- Give an intuitive explanation for polynomial quotient ring, or polynomial ring mod kernel
- Construction of matrices under ZFC axioms
- How to show that $(a+b)^p\le 2^p (a^p+b^p)$
- How to get the adjacency matrix of the dual of $G$ without pen and paper?
- How do you explain the concept of logarithm to a five year old?
- Anticommutative operation on a set with more than one element is not commutative and has no identity element?
- “Natural” example of cosets
- Applications of algebraic topology
- Aunt and Uncle's fuel oil tank dip stick problem
- Is $\mathbb Q \times \mathbb Q $ a denumerable set?
- Solving base e equation $e^x – e^{-x} = 0$
- Subset of a finite set is finite