mathkaveli

When solving non-homogeneous linear differential equations, finding the particular solution (often written as y_p) is the most critical step. You generally have two tools in your belt: the Method of Undetermined Coefficients and the Method of Variation of Parameters. While both aim for the same result, they operate on very different logic. Here is a breakdown of how they stack up. 1. Undetermined Coefficients: The "Educated Guess" This method is the "shortcut" of the differential equation world. It relies on the fact that for certain types of functions, the derivative looks remarkably like the original function.  * How it works: You assume y_p takes the same form as the non-homogeneous term g(x). If g(x) is an exponential like e^(2x), you guess A * e^(2x). If g(x) is sin(x), you guess A * cos(x) + B * sin(x).  * The Constraint: It only works if g(x) is a polynomial, exponential, sine/cosine, or a product/sum of these.  * The Big Pro: It is algebraically straightforw...

Why We Should Never Tell Students What They Can Tell Us A student-centered approach to deeper mathematical thinking There is a deceptively simple idea in teaching that can fundamentally reshape classroom practice: never say anything a student can say. This principle comes from Steve Reinhart’s article Never Say Anything a Kid Can Say! (2000), and it challenges teachers to resist the urge to explain, clarify, or summarize when students themselves are capable of doing that intellectual work. When we connect Reinhart’s insight with Robert Kaplinsky’s instructional reflection and Dan Finkel’s TEDx talk, Five Principles of Extraordinary Math Teaching , a coherent vision of student-centered learning emerges. In that vision, students are active sense-makers and communicators, and the teacher’s primary job is to design experiences that make student thinking visible. Lead with a question Dan Finkel argues that extraordinary math teaching begins with questions worth thinking about, not ...

Carl Friedrich Gauss once wrote that “Mathematics is the queen of the sciences and number theory is the queen of mathematics.” Every time I teach or revisit introductory number theory, I feel the truth of that line settle in. There is something dignified, almost royal, about working with the integers—these simple, familiar numbers that somehow hide endless surprises and patterns. In this post, I want to walk through the major ideas you would encounter in a first number theory course for math majors, and share a bit of the joy behind each topic. After all, number theory isn’t just a subject to study; it’s a subject to savor. Number theory begins with **divisibility**, and even the most basic idea that “a divides b” if b can be written as a times some integer. It feels small at first, but suddenly this tiny definition opens up entire landscapes. The Euclidean Algorithm, for example, takes two integers and dances backward through divisions to reveal the greatest common divisor. There is ...

Every November 5th, the night sky glows with bonfires and fireworks across the United Kingdom. Children recite the familiar words: Remember, remember the Fifth of November, Gunpowder, treason, and plot. I see no reason why gunpowder treason Should ever be forgot. This short rhyme—known as the Bonfire Prayer —dates back over four centuries to the failed Gunpowder Plot of 1605 , when a group of English Catholics, including Guy Fawkes , attempted to blow up the Houses of Parliament and assassinate King James I. Their goal was to end Protestant rule and restore a Catholic monarchy. The plan failed, and Fawkes was captured, tortured, and executed. From Thanksgiving to Tradition In the years that followed, November 5th became known as Guy Fawkes Night or Bonfire Night . The government declared it a national day of thanksgiving for the King’s survival. Communities lit bonfires, rang church bells, and later burned effigies of Guy Fawkes in public squares. The annual celebrati...

In this post, we will explore an elegant mathematical idea that links geometry, calculus, and physics: the Legendre transform (sometimes called the Legendre dual ). The basic concept is that a smooth curve or function can be re-described entirely in terms of the slopes of its tangent lines. Even more remarkably, this process can be reversed to recover the original function. The same idea appears in classical mechanics and thermodynamics as a tool for switching between different physical variables, such as velocity and momentum or entropy and temperature. 1. Geometric Intuition with Tangent Lines Suppose we have a smooth, upward-curving function f(x) . At any point x = a , we can draw a tangent line. The tangent line can be written in slope-intercept form: y = p·x + b Here, p = f′(a) is the slope of the tangent line, and b is its y-intercept, which can be found by substituting x = 0 : b = f(a) − a·f′(a) So, every point a on the curve corresponds to a line with slope p a...

The Hidden Connections Between Totient, Sigma, Tau, Möbius, and Dirichlet Convolution In number theory, certain functions show how integers interact with their divisors. This post introduces five of them — the Euler totient function phi(n), the sum of divisors sigma(n), the number of divisors tau(n), the Möbius function mu(n), and the Dirichlet convolution — using one example number: Example number: n = 12 1) Euler’s Totient Function φ(n) Idea: phi(n) counts how many numbers from 1 to n are coprime to n (that is, share no common factors with n except 1). Formula: If n = p1^a1 * p2^a2 * ... * pk^ak, then phi(n) = n * (1 - 1/p1) * (1 - 1/p2) * ... * (1 - 1/pk) Example with n = 12: 12 = 2^2 * 3^1 phi(12) = 12 * (1 - 1/2) * (1 - 1/3) = 12 * (1/2) * (2/3) = 4 The numbers 1, 5, 7, and 11 are coprime to 12, so phi(12) = 4. 2) The Sigma Function σ(n) Idea: sigma(n) is the sum of all positive divisors of n. Formula: If n = p1^a1 * p2^a2 * ... * pk^a...