Essential Mathematical Methods For Physicists Pdf

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• Chapter 8

  Differential Equations

  8.1 Introduction

  In physics, the knowledge of the force in an equation of motion usually leadsto a differential equation, with time as the independent variable, that gov-erns dynamical changes in space. Almost all the elementary and numerousadvanced parts of theoretical physics are formulated in terms of differentialequations. Sometimes these are ordinary differential equations in one variable(ODE). More often, the equations are partial differential equations (PDE) incombinations of space and time variables. In fact, PDEs motivate physicistsinterest in ODEs. The term ordinary is applied when the only derivativesdy/dx, d2 y/dx2, . . . are ordinary or total derivatives. An ODE is first order ifit contains the first and no higher derivatives of the unknown function y(x),second order if it contains d2 y/dx2 and no higher derivatives, etc.

  Recall from calculus that the operation of taking an ordinary derivative isa linear operation (L)1

  d(a(x) + b(x))dx

  = addx

  + b ddx

  .

  In general,

  L(a + b) = aL() + bL(), (8.1)where a and b are constants. An ODE is called linear if it is linear in theunknown function and its derivatives. Thus, linear ODEs appear as linearoperator equations

  L = F,

  1We are especially interested in linear operators because in quantum mechanics physical quantitiesare represented by linear operators operating in a complex, infinite dimensional Hilbert space.

  410

• 8.2 First-Order ODEs 411

  where is the unknown function or general solution, the source F is a knownfunction of one variable (for ODEs) and independent of , and L is a linearcombination of derivatives acting on . If F = 0, the ODE is called inho-mogeneous; if F 0, the ODE is called homogeneous. The solution of thehomogeneous ODE can be multiplied by an arbitrary constant. If p is a par-ticular solution of the inhomogeneous ODE, then h = p is a solutionof the homogeneous ODE because L( p) = F F = 0. Thus, the gen-eral solution is given by = p + h. For the homogeneous ODE, any linearcombination of solutions is again a solution, provided the differential equationis linear in the unknown function h; this is the superposition principle.We usually have to solve the homogeneous ODE first before searching forparticular solutions of the inhomogeneous ODE.

  Since the dynamics of many physical systems involve second-order deriva-tives (e.g., acceleration in classical mechanics and the kinetic energy operator,2, in quantum mechanics), differential equations of second order occurmost frequently in physics. [Maxwells and Diracs equations are first order butinvolve two unknown functions. Eliminating one unknown yields a second-order differential equation for the other (compare Section 1.9).] Similarly, anyhigher order (linear) ODE can be reduced to a system of coupled first-orderODEs.

  Nonetheless, there are many physics problems that involve first-orderODEs. Examples are resistanceinductance electrical circuits, radioactivedecays, and special second-order ODEs that can be reduced to first-orderODEs. These cases and separable ODEs will be discussed first. ODEs ofsecond order are more common and treated in subsequent sections, involvingthe special class of linear ODEs with constant coefficients. The impor-tant power-series expansion method of solving ODEs is demonstrated usingsecond-order ODEs.

  8.2 First-Order ODEs

  Certain physical problems involve first-order differential equations. Moreover,sometimes second-order ODEs can be reduced to first-order ODEs, which thenhave to be solved. Thus, it seems desirable to start with them. We consider heredifferential equations of the general form

  dy

  dx= f (x, y) = P(x, y)

  Q(x, y). (8.2)

  Equation (8.2) is clearly a first-order ODE; it may or may not be linear, althoughwe shall treat the linear case explicitly later, starting with Eq. (8.12).

  Separable Variables

  Frequently, Eq. (8.2) will have the special form

  dy

  dx= f (x, y) = P(x)

  Q(y). (8.3)

• 412 Chapter 8 Differential Equations

  Then it may be rewritten as

  P(x)dx + Q(y)dy = 0.Integrating from (x0, y0) to (x, y) yields x

  x0

  P(X)dX + y

  y0

  Q(Y)dY = 0. (8.4)

  Here we have used capitals to distinguish the integration variables from theupper limits of the integrals, a practice that we will continue without furthercomment. Since the lower limits x0 and y0 contribute constants, we may ignorethe lower limits of integration and write a constant of integration on the right-hand side instead of zero, which can be used to satisfy an initial condition. Notethat this separation of variables technique does not require that the differentialequation be linear.

  EXAMPLE 8.2.1 Radioactive Decay The decay of a radioactive sample involves an eventthat is repeated at a constant rate . If the observation time dt is small enoughso that the emission of two or more particles is negligible, then the probabilitythat one particle is emitted is dt, with dt 1. The decay law is given by

  dN(t)dt

  = N(t), (8.5)where N(t) is the number of radioactive atoms in the sample at time t. ThisODE is separable

  dN/N = dt (8.6)and can be integrated to give

  ln N = t + ln N0, or N(t) = N0et, (8.7)where we have written the integration constant in logarithmic form for con-venience; N0 is fixed by an initial condition N(0) = N0.

  In the next example from classical mechanics, the ODE is separable but notlinear in the unknown, which poses no problem.

  EXAMPLE 8.2.2 Parachutist We want to find the velocity of the falling parachutist as afunction of time and are particularly interested in the constant limiting ve-locity, v0, that comes about by air resistance taken to be quadratic, bv2, andopposing the force of the gravitational attraction, mg, of the earth. We choosea coordinate system in which the positive direction is downward so that thegravitational force is positive. For simplicity we assume that the parachuteopens immediately, that is, at time t = 0, where v(t = 0) = 0, our initialcondition. Newtons law applied to the falling parachutist gives

  mv = mg bv2,where m includes the mass of the parachute.

• 8.2 First-Order ODEs 413

  The terminal velocity v0 can be found from the equation of motion as t , when there is no acceleration, v = 0, so that

  bv20 = mg, or v0 =

  mg/b.

  The variables t and v separate

  dv

  g bmv2= dt,

  which we integrate by decomposing the denominator into partial fractions.The roots of the denominator are at v = v0. Hence,(

  g bm

  v2)1

  = m2v0b

  (1

  v + v0 1

  v v0

  ).

  Integrating both terms yields v dVg bmV 2

  = 12

  m

  gbln

  v0 + vv0 v = t.

  Solving for the velocity yields

  v = e2t/T 1

  e2t/T + 1v0 = v0sinh tTcosh tT

  = v0 tanh tT ,

  where T =

  mgb is the time constant governing the asymptotic approach of

  the velocity to the limiting velocity v0.Putting in numerical values, g = 9.8 m/sec2 and taking b = 700 kg/m,

  m = 70 kg, gives v0 =

  9.8/10 1 m/sec, 3.6 km/hr, or 2.23 miles/hr, thewalking speed of a pedestrian at landing, and T =

  mbg = 1/

  10 9.8 0.1

  sec. Thus, the constant speed v0 is reached within 1 sec. Finally, because itis always important to check the solution, we verify that our solutionsatisfies

  v = cosh t/Tcosh t/T

  v0

  T sinh

  2 t/T

  cosh2 t/T

  v0

  T= v0

  T v

  2

  Tv0= g b

  mv2,

  that is, Newtons equation of motion. The more realistic case, in which theparachutist is in free fall with an initial speed vi = v(0) = 0 before theparachute opens, is addressed in Exercise 8.2.16.

  Exact Differential Equations

  We rewrite Eq. (8.2) as

  P(x, y)dx + Q(x, y)dy = 0. (8.8)This equation is said to be exact if we can match the left-hand side of it to adifferential d,

  d = x

  dx + y

  dy. (8.9)

• 414 Chapter 8 Differential Equations

  Since Eq. (8.8) has a zero on the right, we look for an unknown function(x, y) = constant and d = 0.

  We have [if such a function (x, y) exists]

  P(x, y)dx + Q(x, y)dy = x

  dx + y

  dy (8.10a)

  and

  x= P(x, y),

  y= Q(x, y). (8.10b)

  The necessary and sufficient condition for our equation to be exact is that thesecond, mixed partial derivatives of (x, y) (assumed continuous) are inde-pendent of the order of differentiation:

  2

  yx= P(x, y)

  y= Q(x, y)

  x=

  2

  x y. (8.11)

  Note the resemblance to Eq. (1.124) of Section 1.12. If Eq. (8.8) correspondsto a curl (equal to zero), then a potential, (x, y), must exist.

  If (x, y) exists then, from Eqs. (8.8) and (8.10a), our solution is

  (x, y) = C.

  We may construct (x, y) from its partial derivatives, just as we construct amagnetic vector potential from its curl. See Exercises 8.2.7 and 8.2.8.

  It may well turn out that Eq. (8.8) is not exact and that Eq. (8.11) is notsatisfied. However, there always exists at least one and perhaps many moreintegrating factors, (x, y), such that

  (x, y)P(x, y)dx + (x, y)Q(x, y)dy = 0

  is exact. Unfortunately, an integrating factor is not always obvious or easyto find. Unlike the case of the linear first-order differential equation to beconsidered next, there is no systematic way to develop an integrating factorfor Eq. (8.8).

  A differential equation in which the variables have been separated is auto-matically exact. An exact differential equation is not necessarily separable.

  Linear First-Order ODEs

  If f (x, y) in Eq. (8.2) has the form p(x)y + q(x), then Eq. (8.2) becomesdy

  dx+ p(x)y = q(x). (8.12)

  Equation (8.12) is the most general linear first-order

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