ENGG2013 Unit 23 First-order Differential Equations Apr, 2011.

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, and some initial condition x(0) = x 0, find a function x(t) satisfying the differential equation and the initial condition. kshum3 A function in x’, x and t

Discharging a capacitor through a resistor kshum4 Initial voltage across the capacitor is V 0. 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) V0V0 + i(t) V’(t) = –V/(RC) V(0) = V 0. Autonomous first-order DE

Population model for bacteria kshum5 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

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. kshum6 Proportionality constant Autonomous first-order DE

Phase line for x’ = kx k>0 k<0 kshum7 x  Phase line 0 x  Unstable equilibrium x  Phase line 0 x  Stable equilibrium

The parameter a in x’ = ax Three types of behaviour – a > 0, exponential growth – a = 0, constant solution – a < 0, exponential decrease kshum8 a=0 a<0 a>0

General Solution to x’=kx Each function of the form f(t) = C e kt is a solution to x’ = k x. Easy to verify f’(t) = (C e kt )’ = C (e kt )’ = C (k e kt ) = k f(t)

Deriving the general solution by power series Suppose we do not know that exponential function is a solution to x’=kx. We can derive it using power series method. Suppose that the solution is a power series in the form c 0 +c 1 t+c 2 t 2 +…, where c 0, c 1, c 2,… are constants to be determine. Assume that we can differentiate term-by-term (c 0 +c 1 t+c 2 t 2 +c 3 t 3 +…)’ = c 1 + 2c 2 t + 3c 3 t 2 + … By comparing like term, c 1 + 2c 2 t + 3c 3 t 2 + … = k(c 0 + c 1 t + c 2 t 2 + …) – c 1 = k c 0 – c 2 = k c 1 /2 = k 2 c 0 /2 – c 3 = kc 2 /3 = k 3 c 0 /3! – in general, c n = k n c 0 /n! kshum10 General solution to x’=kx

Iodine-131 The half life of Iodine-131 is about 8 days. Suppose that is one Becquerel (Bq) of Iodine- 131 initially. Find the number of days until the radioactivity level drop to 0.01 Bq. Let x(t) be the radioactivity level on day t. Initial condition x(0)=1. kshum11 Typical application Unknown parameter

Solution General solution x(t) = C e – t. – Need to determine unknown constants C and. x(0) = 1  C=1. x(8) = 0.5  0.5 = e –8  = Therefore, x(t) = e – t. Solve 0.01 = x(t) = e – t.  t  53 days. kshum12

CLASSIFICATION OF FIRST-ORDER DIFFERENTIAL EQUATIONS kshum13

Nomenclatures “First-order”: only the first derivative is involved. “Autonomous”: the independent variable does not appear in the DE “Linear”: – “Homogeneous” – “Non-homogeneous” c(t) not identically zero kshum14

Examples kshum15 First-order Autonomous Linear Homogeneous Non-homogeneous

Falling body with linear air friction The air resistance is in the direction opposite to that of the motion. The retarding force is directly proportional to v , where v stands for the speed, and  is a constant between 1 and 2. – Slow speed:  =1. – High speed:  =2. kshum16 m Positive direction mg kv Suppose speed is slow and  =1. Linear non-homogeneous g  –10 m/s 2 k>0

RL in series Physical laws – Voltage drop across resistor = V R (t) = R I(t) – Voltage drop across inductor = V L (t) = L I’(t) kshum17 Linear non-homogeneous Current From Kirchhoff voltage law V R (t) + V L (t) = 0.5 sin(  t)

RC in series Physical laws – Voltage drop across resistor = V R (t) = R I(t) – Voltage drop across inductor = C V C (t) = Q(t) kshum18 Charge From Kirchoff voltage law V C (t) + V R (t) = sin(  t) Linear non-homogeneous