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3296Linear ODE with Constant ForcingSolve x'(t)+2x(t)=6 with x(0)=1, and simplify the solution.数学中等derivation未尝试面试订阅3301Logistic Growth Carrying CapacityConsider the logistic ODE x'(t)=1x(t) (1- x(t) 10 ) with initial value x(0)=2. What is the long-run limit of x(t) as t ?数学简单derivation未尝试面试订阅3311Stable Equilibrium of x(1-x)Consider x'(t)=x(1-x). Which equilibrium is stable?数学中等derivation未尝试面试订阅3315Stable Root in a Shifted Logistic FlowConsider x'(t)=(x-1)(3-x). Which equilibrium is stable?数学中等derivation未尝试面试订阅3331Which Fourier Mode Decays More Slowly?Under the heat equation on 0<x< , which survives longer: the mode \sin x or the mode \sin(3x)?数学中等derivation未尝试面试订阅3332Relative Decay of the Second Mode to the FirstIf two heat modes start with the same amplitude, \sin x and \sin(2x), what multiplicative factor relates the amplitude of the second to the first after time t?数学中等derivation未尝试面试订阅3333Same Mode, Larger Diffusion CoefficientFor the same initial mode \sin(2x), which system cools faster: \kappa=1 or \kappa=3?数学中等derivation未尝试面试订阅3334Long-Run Dominant Mode in a Two-Mode MixtureIf the initial condition is u(x,0)=5\sin x+\sin(4x) with zero boundary conditions, which mode dominates at long horizons?数学中等derivation未尝试面试订阅3335Time Needed for a Mode to Shrink by a Factor e^-1For the mode \sin(3x) evolving under u t=2u xx , after what time does its amplitude shrink by a factor of e -1 ?数学中等derivation未尝试面试订阅3336Why High-Frequency Modes Die FirstWhy does the heat equation smooth out small-scale wiggles faster than large-scale structure?数学中等essay未尝试面试订阅3337Maximum Principle IntuitionWhy can the heat equation not spontaneously create a new interior maximum if there is no internal heat source?数学中等essay未尝试面试订阅3338Boundary Conditions MatterWhy do boundary conditions qualitatively matter for heat diffusion on a finite interval?数学中等essay未尝试面试订阅3339Diffusion as Local AveragingExplain in words why the heat equation can be viewed as repeated local averaging.数学中等essay未尝试面试订阅3340Why Diffusion Forgets Fine Initial DetailsWhy does long-time heat evolution remember coarse mass distribution more than fine initial details?数学中等essay未尝试面试订阅3356Feasible Center Means Zero MultiplierConsider minimizing (x-1) 2+(y-1) 2 subject to x+y\ge 1. At the optimum, is the inequality active or inactive, and what does KKT imply about ?数学中等derivation未尝试面试订阅3358Offset Point Already FeasibleConsider minimizing (x-2) 2+(y--1) 2 subject to x+y\ge 0. At the optimum, is the inequality active or inactive, and what does KKT imply about ?数学中等derivation未尝试面试订阅3360High y Coordinate Makes the Constraint SlackConsider minimizing (x--1) 2+(y-3) 2 subject to x+y\ge 1. At the optimum, is the inequality active or inactive, and what does KKT imply about ?数学中等derivation未尝试面试订阅3361Why Convexity Makes KKT So PowerfulWhy do KKT conditions become sufficient, not just necessary, in many convex optimization problems?数学中等essay未尝试面试订阅3362Complementary Slackness as an Economic StatementExplain complementary slackness in plain language to a PM who thinks of constraints as scarce resources.数学中等essay未尝试面试订阅3363When KKT Is Not Enough by ItselfGive one reason why solving the KKT equations in a nonconvex problem may fail to identify the global optimum.数学中等essay未尝试面试订阅