Differential and Integral Calculus Vol. II [2]

Table of contents :
in
DIFFERENTIAL
INTEGRAL
CALCULUS
C-t», —
£= V^+(af)
S-7 «
*',=sinh (■j+c*)
y’ = f(x.y) (1')
y'=f(x, y)
y |*=*, = I/o
d£=f(x, y) (H
C> (3)
i=fA*)h(y) (O

(3)
(1)
/(*. y)=f(i>i)
y* ï)=0
(2)
«M*. y> C) = 0
(4)
(5)
ta"»-T!rlbr
ÈL
£_i
0(n) = /(■*. y, y’, ..., 0*"-»)
p_il±Æ*
y'=îhî(l-x)dxs=ij(lx-t)
%=p- Then S=i-
ÿ=-L VT+p*
in{p+/T+7) =-+Ct
p=sinh('zr+c‘)
(3)
= ±Kcl-sr,/»
■f
Y
sin -nrsmJLw-
FT=T- Yï*‘
y" = f(x, y, y')
(3)
y[ y2
W (ÿi. yt) =
y[ y\
W'x (ylt y2) = (y^—ylyA' = y^l—yly*
W'+a.W^O (6)
r0=c
(r),=,.-c=o
ÿx=p = 0, y;_ 3-0
=*H:
^=1, 2
k* + q = 0, q> 0
k\ + pk1 + q = Q
y = ÿ+y* (3)
CJ = Ti(*). C; =

Citation preview

N. PISKUNOV

differential and intégral calculus VO i n MIR PUBLISHERS • M OSCOW

ABOUT THE BOOK

Textbook by the late Prof. Nikolai Piskunov, D. Sc. (Phys, and Math.), is devoted to the most important divisions of higher mathematics. This édition, revised and enlarged, is published in two volumes, the second volume dealing with the following topics: Differential Equations, Multiple Intégrais, Line and Surface Intégrais, Sériés, Fourier Sériés, The Equations of Mathematical Physics, Operational Calculus and Certain of Its Applica­ tions, Eléments of the Theory of Probability and Mathematical Statistics, Matrices. There are numerous examples and problems in each section of the course; many of them demonstrate the ties between mathematics and other sciences, making the book a useful aid for self-study. This is a textbook for higher technical schools that has gone through several éditions in Russian and alëo been translated into French and Spanish.

H. C. I l H C K y H O B

Æ H O G E P E H U H A J Ib H O E H H H T E rP A / Ib H O E H CM HCJlEHM fl

TOM II

H3ZIATEJlbCTBO «HAYK A» MOCKBA

N. P IS K U N O V

DIFFERENTIAL AND

INTEGRAL CALCULUS VOL. I I Translatée! from the Russian by George Yankovsky

M I R P U B L IS H E R S MOSCOW

First published 1964 Second printing 1966 Third printing 1969 Second édition 1974 Fourth printing 1981

H a

OH3AUÜCKOM R3blKê

© Engli^h translation, Mir Publishers, 1974

P r in te d

in

ih e

U n io n

o f

S o v ie t

S o c ia lis t

R e p u b lic s

CONTENTS

CHAPTER 1 D1FFERENTIAL EQUATIONS

1.1

Statement of the problem. The équation of motion of a body with résistance of the medium proportional to the velocity. The équation of a c a te n a ry ............................................................................................. 11 1.2 D éfinitions................................................................................................. 14 1.3 First-order differential équations (general notions) .......................... 15 1.4 Equations with separated and separable variables. The problem of the disintegration of radium ............................................................... 20 1.5 Homogeneous first-order é q u a tio n s ........................................................ 24 1.6 Equations reducible to homogeneous é q u a tio n s.................................. 26 1.7 First-order liuear é q u a tio n s ................................................................... 29 1.8 Bernoulli’s é q u a t i o n .............................................................................. 32 1.9 Exact differential éq u atio n s................................................................... 34 1.10 Integrating fa c to r...................................................................................... 37 1.11 The envelope of a familv of c u r v e s .................................................... 39 1.12 Singular solutions of a first-order differential é q u a t i o n ................... 45 1.13 Clairaut’s é q u a tio n ................................................................................... 46 1.14 Lagrange’s équation ............................................................................... 48 1.L5 Orthogonal, and isogonal trajecto ries.................................................... 50 1.16 Higher-order differential équations (fundamentals) .......................... 55 1.17 An équation of the form yW = f ( x ) ..................................................... 56 1.18 Some types of second-order differential équations reducible to firstorder équations. Escape-velocity problem . .......................................... 59 1.19 Graphical method of integrating second-order differential équations 66 1.20 Homogeneous linear équations. Définitions andgeneral properties 68 1.21 Second-order homogeneous linear équations with constant coefficients 75 1.22 Homogeneous linear équations of the nth order with constant coeffi­ cients ........................................................................................................ 80 1.23 Nonhomogeneous second-order linear é q u a tio n s .................................. 82 1.24 Nonhomogeneous second-order linear équations with constant coeffi­ cients ........................................................................................................ 86 1.25 Higher-order nonhomogeneous linear é q u a tio n s.................................. 93 1.26 The differential équation of mechanical v ib r a tio n s .......................... 97 1.27 Free o sc illa tio n s ...................................................................................... 98 1.28 Forced o s c illa tio n s.................................................................................. 102 1.29 Systems of ordinary differential é q u a tio n s ......................................... 106 1.30 Systems of linear differential équations with constant coefficients 111 1.31 On Lyapunov’s theory of s ta b ility ........................................................ 117 1.32 Euler's method of approximate solution of first-order differential équations ................................................................................................. 133 1.33 A différence method for approximate solution of differential équa­ 135 tions based on Taylor's formula. Adams m e th o d .............................. 1.34 An approximate method for integrating Systems of first-order diffe­ rential é qua tions...................................................................................... 142 Exercises on Chapter 1 146

Contents

6

CHAPTER 2 MULTIPLE INTEGRALS

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10

Double in té g r a is ...................................................................... Calculating double in té g ra is................................................................... Calculating double intégrais (c o n tin u e d )............................................. Calculating areas and volumes by means of doubleintégrais . . . The double intégral in polar coordinates............................................. Change of variables in a double intégral(general c a s e )..................... Computing the area of a surface ........................................................ The density distribution of matter and the doubleintégral . . . . The moment of inertia of the area of a plane f i g u r e ...................... The coordinates of the centre of gravity of the area of a plane f i g u r e ........................................................................................................ 2.11 Triple intégrais ...................................................................................... 2.12 Evaluating a triple intégral ............................................................... 2.13 Change of variables in a triple i n t é g r a l ............................................. 2.14 The moment of inertia and the coordinates of the centre of gravity of a solid . . .......................................................................................... 2.15 Computing intégrais dépendent on a p aram eter.................................. Exercises on Chapter 2

158 161 166 172 175 182 187 190 191 196 197 198 204 207 209 211

CHAPTER 3 LINE INTEGRALS AND SURFACE INTEGRALS

3.1 3.2 3.3 3.4

Line intégrais ......................................................................................... Evaluating a line in té g ra l....................................................................... Green’s formula ...................................................................................... Conditions for a line intégral to be independent of the path of inté­ gration .................................................................................................... 3.5 Surface in té g r a is ...................................................................................... 3.6 Evaluating surface in té g ra is................................................................... 3.7 Stokes* formula ...................................................................................... 3.8 Ostrogradsky’s f o r m u la ........................................................................... 3.9 The Hamiltonian operatorand some ap p lic a tio n s................................ Exercises on Chapter 3

216 219 225 227 232 234 236 241 244 247

CHAPTER 4 SERIES

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19

Sériés. Sum of a s é r i é s .......................................................................... Necessary condition for convergence of a sériés . . . . ! ! ! ! ! ! Comparing sériés with positive terms ............................. ! ! ! ! ! ! D’Alembert’s t e s t ...................................................................................... Cauchy’s t e s t ....................................................................... ! ! . ! ! ! ! The intégral test for convergence of a s é r i é s ......................................[ Alternating sériés. Leibniz* th e o re m .....................................................’ Plus-and-minus sériés. Absolute and conditional convergence . . . . Functional s é r i é s .............................................................................. Dcminated s é r i é s ........................................................ ! ! ! ! ! ! ! ! ! The continuity of the sum of as é r i é s .................................... Intégration and différentiation of sériés .......................... * Power sériés. Interval of convergence ............................................ ’ . Différentiation of power sé rié s ........................................................... Seriez in powers o f \ —a ................................. . ! . ! . ! . . ! ! ! Taylor*s sériés and JVu>çlaurin*ssériés ....................................... Sériés expansion of f u n b tio n s ................................................ Euler’s formula .................................................... ! . 1 ! ! The binomial sériés .

253 256 258 260 264 266 269 271 274 275 277 280 283 288 289 290 292 294 295

Contents 4.20 Expansion of the function ln (l-fx ) in a power sériés. Computing lo g a rith m s ................................................................................................. 4.21 Sériés évaluation of definite intégrais . .# ......................................... 4.22 Integrating differential équations by means of sériés ................... 4.23 BesseFs é q u a tio n ...................................................................................... 4.24 Sériés with complex te rm s .................. .................................................... 4.25 Power sériés in a complex v a r i a b l e .................................................... 4.26 The solution of first-order differential équations by the method of successive approximations (methodof ité ra tio n ).................................. 4.27 Proof of the, existence of a solution of a differential equatfon. Error estimation in approximate s o lu tio n s ...................................................... 4.28 The uniqueness theorem of the solution of a differential équation Exercises on Chapter 4

7 297 299 301 303 308 '309 312 313 318 319

CHAPTER 5 FOU R 1ER SERIES

5.1 5.2 5.3 5.4 5.5 5.6 5.7

Définition. Statement of the p ro b le m ........................................... 327 Expansions of functions in Fourier sériés ......................................... 331 A remark on the expansion of a periodic function in aFouriersériés 336 Fourier sériés for even and odd f u n c tio n s .................................... 338 The Fourier sériés for a function with period 2 1 ........................ 339 On the expansion of a nonperiodic function in aFourier sériés . . 341 Mean approximation of a given function by a trigonométrie poly­ nomial ......................................................................................................... 343 5.8 The Dirichlet intégral ........................................................................... 348 5.9 The convergence of a Fourier sériés at a given p o i n t ....................... 351 5.10 Certain sufficient conditions for the convergence of aFouriersériés 352 355 5.11 Practical harmonie a n a ly s is ............................................................. 5.12 The Fourier sériés in complex f o r m .............................................. 356 5.13 Fourier intégral ...................................................................................... 358 5.14 The Fourier intégral in complex f o r m .......................................... 362 5.15 Fourier sériés expansion with respect to an orthogonal System of fun ctio n s..................................................................................................... 364 5.16 The concept of a linear function space. Expansion of functions in Fourier sériéscompared with décomposition of vectors . . . . . . 367 Exercises on Chapter 5 371 CHAPTER 6 EQUATIONS OF MATHEMATICAL PHYSICS

6.1 Basic types of équations of mathematical p h y sics.............................. 6.2 Deriving the équation of the vibrating string. Formulating the boundary-value problem. Deriving équations of electric oscillations in w ires....................................................................... i ................................. 6.3 Solution of the équation of the vibrating string by the method of séparation of variables (the Fourier m e th o d )..................................... 6.4 The équation of heat conduction in a rod. Formulation of the boundary-value problem .............................................................................. 6.5 Heat transfer in space .......................................................................... 6.6 Solution of the first boundary-value problem for the heat-conduction équation by the method of nnite d ifféren ces................................. . 6.7 Heat transfer in an unbounded r o d .................................................... 6.8 Problems that reduce to investigating solutions of the Laplace équa­ tion. Stating boundary-value problem s................................................. 6.9 The Laplace équation in cylindrical coordinates. Solution of the Di­ richlet problem for an annulus with constant values of the desired function on the inner and outer circumferences..................................

374 375 378 382 384 387 389 394 399

Contents

6.10 The solution of Dirichlet’s problem for a circle 6.11 Solution of the Dirichlet problem by the method of finite différences Exercises on Chapter 6 ......................................................................................

.

8

405 407

CHAPTER 7 OPERATIONAL CALCULAIS AND CERTAIN OF ITS APPLICATIONS

7.1 7.2 7.3

The original function and its tr a n s f o r m .................. Transforms of the functions a0 (f). s in /, cos / .................................... The transform of a function with changed scale of the independent variable. Transforms of the functions sin af, cos a t .......................... 7.4 The linearity property of a tr a n s f o r m ................................................. 7.5 The shift theorem .................................................................................. 7.6 Transforms of the functions e~a t, sinh at, cosh a t, e " a / sin a tt e~ai cos a t ................................................................................................. 7.7 Différentiation of transform s................................................................... 7.8 The transforms of d é riv a tiv e s ............................................................... 7.9 Table of transform s.................................................................................. 7.10 An auxiliary équation for a given differential é q u a tio n ................... 7.11 Décomposition th e o r e m ........................................................................... 7.12 Examples of solutions of differential équations and Systems of diffe­ rential équations by the operational m e t h o d ..................................... 7.13 The convoiution th e o r e m ....................................................................... 7.14 The differential équations of mechanical vibrations. The differential équations of electric-circuit th e o ry ........................................................ 7.15 Solution of the differential équation ofo sc illa tio n s ............................ 7.16 Investigating free o sc illa tio n s............................................................... 7.17 Investigating mechanical and electrical oscillations in the case of a periodic external force ........................................................................... 7.18 Solving the oscillation équation in thecase of ré so n a n c e ................. 7.19 The delay th e o re m .................................................................................. 7.20 The delta function and its transform .................................................... Exercises on Chapter 7 ......................

411 413 414 415 416 416 417 419 420 422 426 428 429 432 433 435 435 437 439 440 443

CHAPTER 8 ELEMENTS OF THE THEORY OF PROBABILITY AND MATHEMATICAL STATISTICS

8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14

Random event. Relative frequency of a random event. The probability of an event. The subject of probability th e o ry ....................... The classical définition of probability and the calculation of proba­ bilités ......................................................................................................... The addition of probabilités. Complementary random events . . . Multiplication of probabilités of independent e v e n t s ....................... Dépendent events. Conditional probability. Total probability . . . Probability of causes. Bayes's fo rm u la ................................................. A discrète random variable. The distribution law of a discrète ran­ dom variable .............................................................................................. Relative frequency and the probability of relative frequency in repeated t r i a l s .............................................................................................. The mathematical expectation of a discrète random variable . . . Variance. Root-mean-square (standard) déviation. Moments . . . . Functions of random variables ............................................................ Continuous random variable. Probability density function of a continuous random variable. The probability of the random variable falling in a specified in t e r v a l ................................................................ The distribution function. Law of uniform d istrib u tio n ................... Numerical characteristics of a continuous random variable . . . .

445 447 449 452 454 457 460 462 466 471 474 475 479 482

401

Contents 8.15 Normal distribution. The expectation of a normal distribution . . . 8.16 Variance and standard déviation of a normally distributed random v a r i a b l e .......................................................« .......................................... 817 The probability of a value of the random variable falling in a given interval. The Laplace function. Normal distribution function . . . 8.18 Probable e r r o r .......................................................................................... 8.19 Expressing the normal distribution in terms of the probable error. The reduced Laplace fu n c tio n ................................................................ 8.20 The three-sigma rule. Error d is trib u tio n ............................................. 8.21 Mean arithmetic e r r o r ............................................................................... 8.22 Modulus of précision. Relationships between the characteristics of the distribution of e r r o r s ....................................................................... 8.23 Two-dimensional random v a ria b le s............................. 8.24 Normal distribution in the plane ......................................................... 8.25 The probability of a two-dimensional random variable falling in a rectangle with si des parai lel to the principal axes of dispersion under the normal distribution l a w ........................................................ 8.26 The probability of a two-dimensional random variable falling in the ellipse of dispersion................................................................................... 8.27 Problems of mathematical statistics. Statistical d a t a ....................... 8.28 Statistical sériés. H istogram .................................................................... 8.29 Determining a suitable value of a measured q u a n tity ....................... 8.30 Determining the parameters of a distribution law. Lyapunov’s theorem. Laplace's th e o r e m ........................................................................... Exercises on Chapter 8 .......................................................................................

9 485 487 488 493 494 496 497 498 499 502 504 506 507 508 511 512 516

CHAPTER 9 MATRICES

9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8

Linear transformations.Matrix n o t a t i o n ............................................... General définitions involving m a t r i c e s ................................................. Inverse transformation................................................ Operations on matrices. Addition of m atrices...................................... Transforming a vector into another vector by means of a matrix Inverse m a t r i x .............................................................................. Malrix in v e r s io n ...................................................................................... Matrix notation for Systems oflinear équations and solutions of Systems of linear e a u a tio n s .................................................................... 9.9 Solving Systems of linear équations by thematrix m ethod................. 9.10 Orthogonal mappings. Orthogonal m a tr ic e s ......................................... 9.11 The eigenvector of a linear tran sfo rm atio n......................................... 9.12 The matrix of a linear transformation under which the base vectors are eigenvectors .............................................. * .................................... 9.13 Transforming the matrix of a linear transformation when changing the basis ...................................................................................................... 9.14 Quadratic forms and their transformation .......................................... 9.15 The rank of a matrix. The existence of solutions of a System of linear équations ...................................................................................... 9.16 Différentiation and intégration of m a tr ic e s ......................................... 9.17 Matrix notation for Systems of differential équations and solutions of Systems of differential équations with consta/it coefficients . . . 9.18 Matrix notation for a linear équation oforder n ................................ 9.19 Solving a System of linear differential équations with variable co­ efficients by the method of successive approximations using matrix n o t a t i o n ..................................................................................................... Exercises on Chapter 9 ...................................................................................... a p p e n d i x ......................................................................................................... i n d e x .................................................................................................................

519 522 524 526 529 531 532 534 535 537 540 543 544 547 549 550 552 557 558 563 565 567

CHAPTER

1

D1FFERENTIAL EQUATIONS

1.1

STATEMENT OF THE PROBLEM. THE EQUATION OF MOTION OF A BODY WITH RESISTANCE OF THE MEDIUM PROPORTIONAL TO THE VELOCITY. THE EQUATION OF A CATENARY

Let a function y = f(x) reflect the quantitative aspect of some phenomenon. Frequently, it is not possible to establish directly the type of dependence of y on x t but it is possible to give the relation between the quantities x and y and the dérivatives of y with respect to x: y \ y \ . . . , yin). That is, we are able to Write a differential équation. From the relationship established between the variables jt, y and the dérivatives it is required to détermine the direct depen­ dence of y on x\ that is, to find y = f(x) or, as we say, to integrate the differential équation. Let us consider two examples. Example 1. A body of mass m is dropped from some height. It is required to establish the law according to which the velocity v will vary as the body falls, if, in addition to the force of gravity, the body is acted upon by the decelerating force of the air, which is proportional to the velocity (with con­ stant of proportionality k)\ in other words, it is required to find v = f( t) . Solution. By Newton’s second law

where

is the accélération of a moving body (the dérivative of the velocity

with respect to time) and F is the force acting on the body in the direction of motion. This force is the résultant of two forces: the force of gravity mg and the force of air résistance,— kv, which has the minus sign because it is in the opposite direction to that of the velocity. And so we hâve m ^ = m g — kv

(1)

This relation connects the unknown function v and its dérivative

we hâve at a differential équation in the unknown function v. This is the équation of mo­ tion of certain types of parachutes. To solve a differential équation means to find a function v = f( t) such that identically satisfies the given differential équation. There is an infinitude of such functions. The student can easily verify that any function of the form ( 2)

12

Ch. 1 Differential Equations

satisfies équation (1) no matter what the constant C is. Which one of ttiese functions yields the sought-for dependence of v o n /?T o find it we take advantage of a supplementary condition: when the body was dropped it was imparted an initial velocity v0 (which may be zéro as a particular case); we assume this initial velocity to be known. But then the unknown function v = f(t) must be such that when t = 0 (when motion begins) the condition v = v 0 is fulfilled. Substituting t = 0, v = u0 into formula (2), we find

whence -o , C -t»

—

Thus, the constant C fs found, and the sought-for dependence of v on / is

From this formula it follows that for sufficiently large t the velocity v dé­ pends but slightly on u0. It will be noted that if k = 0 (the air résistance is absent or so small that we can disregard it), then we hâve a resuit familiar from physics: * v = v0+ g t (2") This function satisfies the differential équation (1) and the initial condition: v = v0 when t = 0. Example 2. A flexible homogeneous thread is suspended at two ends. Find the équation of the curve that it describes under its own weight (it is the same as for any suspended ropes, wires, chains). Solution. Let Af0 (0, b) be the lowest point of the thread, and M an arbitrary point (Fig. 1). Let us consider a part of the thread M 0M. This part is in equilibrium under the action of the résultant of tnree forces: (1) the tension 7\ acting along the tangent at the point M and forming an angle

) = 0 or f* y *ny \ _ a ’ dx* ’ • • • ’ dxn J

If the sought-for function y = f{x) is a function of one indepen­ dent variable, then the differential équation is called ordinary. In this chapter we shall deal only with ordinary differential équations. * Définition 2. The order of a differential équation is the order of the highest dérivative which appears. For example, the équation y '— 2*ÿa + 5 = 0 is an équation of the first order. The équation y"-\-ky'—by—sinx = 0 is an équation of the second order, etc. The équation considered in the preceding section in Example 1 is an équation of the first order, in Example 2, one of the second order. Définition 3. The solution or intégral of a differential équation is any function y = f(x), which, when put into the équation, converts it into an identity. * ln addition to ordinary differential équations, mathematical analysis also deals with partial differential équations. Such an équation is a relation between an unknown function z that is dépendent upon twoorseveral variables*, y ........ these variables x, y, . . . . and the partial dérivatives of z:

^

, etc.

The following is an example of a partial differential équation in the unknown function z(x, y): dz dz X Fx= y Ty It is easy to verify that this équation is satisfied by the function z = x 2y2 (and also by a multitude of other functions). In this course partial differential équations are discussed in Chapter 6.

1.3

First’Order Differentiai Equations

15

Example 1. Suppose we hâve the équation

The functions y = sin*, y = 2 cos x, t/ = 3 sin x — cos x and, in general, functions of the form ^ C x s i n x , y = C2 cos x or y = Ct sin jc + C2 cos x are solutions of the given équation for any choice of constants Cx and Ca; this is évident if we put these functions into the équation. Example 2. Let us consider the équation y'x — x2— y = 0 Its solutions are ail functions of the form y = x* + Cx where C is any constant. Indeed, differentiating the functions y = x®+ Cx, we find y'=*2x + C Putting the expressions for y and y ' into the initial équation, we get the identity (2x + C ) x - x 2- x 2- C x = 0 Each of the équations considered in Examples 1 and 2 has an infinitude of solutions. 1.3

F1RST-ORDER DIFFERENTIAL EQUATIONS (GENERAL NOTIONS)

1. A differential équation of the first order is of the form F(x, y, y') = 0 (1) If this équation can be solved for y', it can be written in the form y’ = f(x.y)

(1')

In this case we say that the differential équation is solved for the dérivative. For such an équation the following theorem, called the theorem of existence and uniqueness of solution of a differen­ tial équation, holds. Theorem. If in the équation y'= f(x, y)

the function f(x, y) and its partial dérivative with respect to y, ■gj, are continuons in some domain D, in an xy-plane, containing sortie point (x0, y0), then there is a unique solution to this équation, t/ =

]. The condition that for x = x0 the function y must be equal to the given number yQis called the initial condition. It is frequently written in the form y |*=*, = I/o Définition 1. The general solution of a first-order differential équation is a function y = (f(x, C) (2) which dépends on a single arbitrary constant C and satisfies the following conditions: (a) it satisfies the differential équation for any spécifie value of the constant C; (b) no matter what the initial condition y —y0 for x = x0, that is, (y)x=x. — ÿo> it is possible to find a value C = C„ such that the function y — ip(x, C0) satisfies the given initial condition. It is assumed here that the values x0 and y0 belong to the range of the variables x and y in which the conditions of the existence and uniqueness theorem are fulfilled. 2. In searching for the general solution of a differential équation we ofien arrive at a relation like ) = 0 j

or, which is the same thing, by eliminating C from the équations y = xC 4- ib (C) * + tc (Q = 0

Thus, the singular solution of Clairaut’s équation defines the envelope of a family of straight lines represented by the complété intégral (4). Example. Find the general and singular solutions of the équation dy a dx dy y=x 2 dx

/■+(=)

4a

Ch. 1 Differential Equations Solution. The general solution is obtained by substituting C for y = xC-

aC FT +c5

To obtain the singular solution, differentiate the latter équation with res­ pect to C:

(1 + C2) 2 The singular solution (the équation of the envelope) is obtained in parametric form (where the parameter is C):

(1 + C 2) 2 aC3 3

(1 + C2) 2 Eliminating C, we get a direct relationship between x and y Raising both sides of each equa2 tion to the power and adding the résultant équations termwise, we get the singular solution in the following form: 2

_2_

2

x 3 + y, 33 = a '3 This is an astroid. However, the envelope of the family (and, hence, the singular solution) is not the entire astroid, but only its left half (since it is évident from the parametric équations that jt< ;0) (Fig. 15). 1.14

LAGRANGE’S EQUATION

The Lagrange équation is an équation of the form

y = xy W )+ W )

( 1)

where

' (p) +♦' (p)] %

(O

From this équation we can straightway find certain solutions: namely, it becomes an identity for any constant value p = p„ that satisfies the condition Po—

Separating variables, we get (assuming, for the time being, that s 7^ s0)

FT=T-

Yï*‘

Again we consider that s = 0 when / = 0 . Integrating this équation, we get

.2 .2

f l

(8 ')

or whence (9) We take the plus sign in front of the root. From the note at the end of the solution it follows that there is no need to consider the case with the mi­ nus sign.

1.18 Second-Order Equations Reducible ta First-Order Equations

65

Note. When solving, we assumed that s ^ s0» But it is clear,. by direct sub­ stitution , that the fonction (9) is the solution of équation (6 ') for any value of t. Let it be recalled that the solution (9) is an ^pproximate solution of équa­ tion (5), since équation (6) was replaced by the approximate équation (6 '). Equation (9) shows that the point M (which may be regarded as the extremity of the pendulum) performs harmonie oscillations with a period T — 2n This period is in dépendent of the amplitude s0. Example 4. Escape-velocity probfem. Détermine the smallest velocity with whfch a body mnst be thrown vertically upwards so that it wi11 not return to the earth. Air resistar.ee is neglectedi. Solution. Dénoté the mass of the earth and the mass of the body by M and m respectively. By Newton’s Iaw of gravitation, the force of attraction f acting on the body m is

where r is the distance between the centre of the earth and the centre of gravity of the body, and k is the gravitational constant. The differential équation of motion of this body with mass m will be d2r M •m

md T '= - k ~ W or A* ~ * r2 The minus sign indicates that the accélération is négative. The differential équation (10) is an équation of type (2). We shall solve it for the following initial conditions: for / = 0

r = R,

^=*V

Here, R is the radius of the earth and v0 is the launching velocity. We dénoté dr_ d2r _dv _dv dr dv d t ~ V> dt2~~dt~~dr " dt~~Vdr where v is the velocity of motion. Putting this into (10), we get M dv k dr r* Separating variables, we obtain

vdv = —kM —• Intégrâting this équation, we find v2 1 -= k M ^ C t

(«O

From the condition that v — v0 at the earth’s surface (for r = R)f we détermine Cx* vô

Ch. 1 Differential Équations

66

We put the value of Cx into (11):

or

It is given that the body should move so that the velocity is always positive; v2 kM > 0 . Since for a boundless increase of r the quantity — becomes

hence,

v2 arbitrarily small, the condition -y > 0 will be fulfilled for any r only for the case (13) or

Hence, the lowest velocity will be .. .a n lined by the équation (14) where k = 6 .66 - 1 0 - 8 cm3/g-sec2 R = 63-107 cm At the earth’s surface, for r = /?, the accélération of gravity isg(g = 981 cm/sec2). For this reason, from (10) we obtain

or

Putting this value of Af into (14), we obtain y0=

Y"2-981 -63-107«a 11.2*10* cm /sec= ll.2 km/sec 1.19 GRAPHICAL METHOD OP INTEGRATING SECOND-ORDER DIFFERENTIAL EQUATIONS t

Let us find out the géométrie meaning of a second-order differential équation. Suppose we hâve an équation y" = f(x, y, y') (i) Dénoté by

so that yx and y2 should be linearly independent. _2x Noting that in our case ai= -j— ~ » we hâve, by ( 10), ç 2x dx

■In II -*«|

+ . . . + 0^ = 0

( 1)

We shall assume that a,, a2, . an are constants. Before giving a method for solving équation (1). we introduce a définition that will be needed later on. D é fin itio n I . If for ali x of the interval [a, b] we hâve the equality