Yesterday· Independent variable, dependent variable andparametersInitial conditions General solutionDirection fieldAutonomous differential equations-PhaselineEquilibrium-Stableandunstable
Yesterday • Independent variable, dependent variable and parameters • Initial conditions • General solution • Direction field • Autonomous differential equations. – Phase line • Equilibrium – Stable and unstable
The initial value problem. Given a differential equation,F(x,x, t) = 0A function in x, x and tand some initial condition x(O) = Xorfind a function x(t) satisfying the differentialequation and the initial condition
The initial value problem • Given a differential equation, and some initial condition x(0) = x0 , find a function x(t) satisfying the differential equation and the initial condition. A function in x’, x and t
Discharging a capacitor through aresistorInitial voltage across thecapacitor is VoThe switch is closed at t = 0Voi(t): Voltage drop is proportional toelectric charge in capacitor.V(t) = Q(t) / CV'(t) = i(t) / CV'(t) = -VI(RC)Voltage drop at resistor isV(0) = Vodirectly proportional tocurrent.Autonomousfirst-orderDEV(t) R = i(t)
Discharging a capacitor through a resistor • Initial voltage across the capacitor is V0 . • The switch is closed at t = 0. • Voltage drop is proportional to electric charge in capacitor. V(t) = Q(t) / C V’(t) = i(t) / C • Voltage drop at resistor is directly proportional to current. V(t) R = i(t) V0 + i(t) V’(t) = –V/(RC) V(0) = V0 . Autonomous first-order DE
Population model for bacteriaBacteria reproduce by binary fissionhttp://en.wikipedia.org/wiki/Bacteria The rate of change of the population P(t) isproportional to the size of population:dP/dt = k Pwhere k is a positive proportionality constant.Autonomousfirst-orderDE
Population model for bacteria • Bacteria reproduce by binary fission. • The rate of change of the population P(t) is proportional to the size of population: dP/dt = k P where k is a positive proportionality constant. Autonomous first-order DE h t t p:/ e n.wikip e dia.o r g / wiki/ B a c t e ria
Radioactive decayhttp://en.wikipedia.org/wiki/Radioactivedecay Radioactive decay is the processby which an atomic nucleus of anunstable atom loses energy by emitting ionizingparticles. The number of decay events is proportional tothe number of atoms present.Let N(t) be the number of radioactive atomat time t.dN(t)-入N(t)Autonomousdtfirst-orderDEProportionalityconstant
Radioactive decay • Radioactive decay is the process by which an atomic nucleus of an unstable atom loses energy by emitting ionizing particles. • The number of decay events is proportional to the number of atoms present. • Let N(t) be the number of radioactive atom at time t. h t t p:/ e n.wikip e dia.o r g / wiki/ R a dio a c tiv e _ d e c a Proportionality y constant Autonomous first-order DE