Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

More documents

Recommendations

Info

334_Differential Equations_[Cfi. 9 whence we find v: and, consequently, the general solution is obtained in the form Find the general integrals of the equations: 2785. -=*. ax x 2786. + = x*. 2787*. (\ 2788. y*dx(2xy Find the particular solutions that satisfy the indicated conditions: 2789. X y' + y e* = Q\ y = b when x = a. 2790. y' j-2-7 2791. y' yianx = ; 1-- * = 0; y = when x-0. cos x */ = when jt = 0. Find the general solutions of the equations: 2792. *l + JL = X m ' y* dx x 2793. 2xy x y 2794. 0dx + ( 2795. 3xdy--=y(l +x sin A: 3y* smx)dx. 2796. Given three particular solutions y, y lt y 2 of a linear equation. Prove that the expression ^^ remains unchanged for any x. What is the geometrical significance of this result? 2797. Find the curves for which the area of a triangle formed by the *-axis, a tangent line and the radius vector of the point of tangency is constant. 2798. Find the equation of a curve, a segment of which, cul off on the x-axis by a tangent line, is equal to the square of the ordinate of the point of tangency. 2799. Find the equation of a curve, a segment of which, cut off on the y-axis by a tangent line, is equal to the subnormal. 2800. Find the equation of a curve, a segment of which, cut off on the y-axis by a tangent line, is proportional to the square of the ordinate of the point of tangency.
Sec. 6] Exact Differential Equations. Integrating Factor 335 2801. Find the equation of the curve for which the segment of the tangent is equal to the distance of the point of intersection of this tangent with the x-axis from the point M (0,a). Sec. 6. Exact Differential Equations. Integrating Factor 1. Exact differential equations. If for the differential equation P(x.y) dx+Q(x t y)dy = Q (1) the equality :p =-^- is fulfilled, then equation (1) may be written in the form dU (x, t/) = and is then called an exact differential equation. The general integral of equation (1) is U (x, y) = C. The function U (x, y) is determined by the technique given in Ch. VI, Sec. 8, or from the formula (see Ch, VII, Sec. 9). Example 1. Find the general integral of the differential equation (3x 2 + 6Af/ 2 ) dx + (6x*y + 4y') dy = 0. Solution. This is an exact differential equation, since --J whence Here, 5= i2X y and, hence, the equation is of the form K/ = 0. and U = (3^ 2 + 6xy*) dx + q> (y) = x> + 3*V + q> (y). Differentiating U with respect to y, we find -y = 6jc f y -f = cp' (y) x l y + 4y* (by hypothesis); from this we get = q> '(//) 4(/* and = q> (y) y* + C*. We finally get s 4 f/(r, jt t/)-= -f-3xV + .V +C C( , consequently, xs -f3^V + / = C is the . sought-for general integral of the equation. 2. Integrating factor. If the left side of equation (l)is not a total (exact) differential and the conditions of the Cauchy theorem are fulfilled, then there exists a function U, = U.(A', y) (integrating factor) such that \i(Pdx+ Qdy) = dU. (2) Whence it is found that the function u, satisfies the equation The integrating factor u, is readily found in two cases:
Page 2 and 3:
TEXT FLY WITHIN THE BOOK ONLY
Page 5:
OUP-43 30-1.71 5,000 OSMANIA UNIVER
Page 8 and 9:
T. C. BapaneHKoe. C. M. Koean, r Jl
Page 10 and 11:
TO THE READER MIR Publishers would
Page 12 and 13:
Contents Sec. 6. Integrating Certai
Page 14 and 15:
8 Chapter Contents Sec. 13. Linear
Page 17 and 18:
Sec. 1. Functions Chapter I INTRODU
Page 19 and 20:
Sec 1] Functions 13 8. Find the rat
Page 21 and 22:
Sec. 1]_Functions_15 34. Prove that
Page 23 and 24:
Sec. 2] Graphs of Elementary functi
Page 25 and 26:
Sec. 2] Graphs of Elementary functi
Page 27 and 28:
Sec. 2] Graphs of Elementary Functi
Page 29 and 30:
Sec. 3] Limits 23 For the existence
Page 31 and 32:
Sec. 3] 179. Hm (Vn + 1 \f~n). n -+
Page 33 and 34:
Sec. #] Limits 27 209. lim 210. lim
Page 35 and 36:
^Sec 3] Limits Therefore, Example 9
Page 37 and 38:
Sec. 31 Limits 31 269. a) 270. lim-
Page 39 and 40:
Sec. 4] Infinitely Small and Large
Page 41 and 42:
Sec. 4] Infinitely Small and Large
Page 43 and 44:
Sec. 5] Continuity of Functions 37
Page 45 and 46:
Sec. 5]_Continuity of Functions_39
Page 47 and 48:
Sec. 5] Continuity of Functions 334
Page 49 and 50:
Sec. 1] Calculating Derivatives Dir
Page 51 and 52:
Sec. 1] Calculating Derivatives Dir
Page 53 and 54:
Sec. 2]__Tabular Differentiation_47
Page 55 and 56:
Sec. 2]_Tabular Differentiation_49
Page 57 and 58:
Sec. 2] Tabular Differentiation_ 51
Page 59 and 60:
Sec. 2] Tabular Differentiation 504
Page 61 and 62:
Sec. 2] Tabular Differentiation 55
Page 63 and 64:
Sec. 3] The Derivatives of Function
Page 65 and 66:
Sec. 3] The Derivatives of Function
Page 67 and 68:
Sec 4] Geometrical and Mechanical A
Page 69 and 70:
Sec. 4\ Geometrical and Mechanical
Page 71 and 72:
Sec. 4] Geometrical and Mechanical
Page 73 and 74:
Sec. 5]_Derivatives of Higher Order
Page 75 and 76:
Sec. 5]_Derivatives of Higher Order
Page 77 and 78:
Sec. 6] Differentials of First and
Page 79 and 80:
Sec. 6}_Differentials of First and
Page 81 and 82:
Sec. 7]_Mean-Value Theorems_75 745.
Page 83 and 84:
Sec. 8] Taylor's Formula 77 765. a)
Page 85 and 86:
Sec. 9]_ V Hospital-Bernoulli Rule
Page 87 and 88:
Sec. 9] L' Hospital-Bernoulli Rule
Page 89 and 90:
Chapter III THE EXTREMA OF A FUNCTI
Page 91 and 92:
Sec. 1] The Extrema of a Function o
Page 93 and 94:
Page 95 and 96:
Page 97 and 98:
Sec. 2]_The Direction of Concavity.
Page 99 and 100:
Sec. 3]_Asymptotes_93 Example 2. Fi
Page 101 and 102:
Sec. 3]_Asymptotes_95 Solution. Sin
Page 103 and 104:
Sec 4] Graphing Functions by Charac
Page 105 and 106:
CM CM -h CO O non ' > C " -~ a - >
Page 107 and 108:
Sec. />] Differential of an Arc. Cu
Page 109 and 110:
Sec 5] Differential of an Arc. Curv
Page 111 and 112:
Sec. 5]_Differential of an Arc. Cur
Page 113 and 114:
Sec. 1. Direct Integration 1. Basic
Page 115:
Sec 1] Direct Integration_109 1048*
Page 119 and 120:
Sec. 2] Integration by Substitution
Page 121 and 122:
Sec. 2] Integration by Substitution
Page 123 and 124:
Sec. 3\_Integration by Parts_H7 Exa
Page 125 and 126:
Sec 4] Standard Integrals Containin
Page 127 and 128:
Sec. 5] Integration of Rational Fun
Page 129 and 130:
Sec. 5] In t egration of Rational F
Page 131 and 132:
Sec. 6] Integrating Certain Irratio
Page 133 and 134:
Sec. 6] Integrating Certain Irratio
Page 135 and 136:
Sec. 7]_Integrating Trigonometric F
Page 137 and 138:
Sec. 7]_Integrating Trigonometric F
Page 139 and 140:
Sec. 8] Integration of Hyperbolic F
Page 141 and 142:
Sec. 11] Using Reduction Formulas 1
Page 143 and 144:
Sec. 12] Miscellaneous Examples on
Page 145 and 146:
Sec. 1] The Definite Integral as th
Page 147 and 148:
Sec. 2]_Evaluating Definite Integra
Page 149 and 150:
Sec. 3]_Improper Integrals_143 Sec.
Page 151 and 152:
Sec. 3] Improper Integrals 145 Solu
Page 153 and 154:
Sec. 4] Change of Variable in a Def
Page 155 and 156:
Sec. 5] Integration by Parts 149 Bu
Page 157 and 158:
Sec. 6] Mean-Value Theorem 151 In p
Page 159 and 160:
Sec. 7] The Areas of Plane Figures
Page 161 and 162:
Sec. 7] The Areas of Plane Figures
Page 163 and 164:
Sec. 7\_The Areas of Plane Figures_
Page 165 and 166:
Sec. 8] The Arc Length of a Curve 1
Page 167 and 168:
Sec, 9] Volumes of Solids 161 1673.
Page 169 and 170:
Sec. 9] Volumes of Solids 163' Solu
Page 171 and 172:
Sec. 9]_Volumes of Solids_165 1693.
Page 173 and 174:
Sec. 10] The Area of a Surface of R
Page 175 and 176:
Sec. II] Moments. Centres of Gravit
Page 177 and 178:
Sec 11] Moments. Centres of Gravity
Page 179 and 180:
&?c. 12] Applying Definite Integral
Page 181 and 182:
Sec. 12] Applying Definite Integral
Page 183 and 184:
Page 185 and 186:
Page 187 and 188:
Sec. 1) Basic Notions [SI Putting *
Page 189 and 190:
Sec. 1]_Basic Notions_183 1790*. Le
Page 191 and 192:
Sec. 3] Partial Derivatives 185 180
Page 193 and 194:
Sec. 4]_Total Differential of a Fun
Page 195 and 196:
Sec. 4}_Total Differential of a Fun
Page 197 and 198:
Sec. 5]_Differentiation of Composit
Page 199 and 200:
Sec. 6]_Derivative in a Given Direc
Page 201 and 202:
Sec. 6] Derivative in a Given Direc
Page 203 and 204:
Sec. 7]_Higher-Order Derivatives an
Page 205 and 206:
Sec. 7] Higher-Order Derivatives an
Page 207 and 208:
Sec. 7]_Higher-Order Derivatives an
Page 209 and 210:
Sec. ti\_Integration of Total Diffe
Page 211 and 212:
Sec. 9]_Differentiation of Implicit
Page 213 and 214:
Sec. 9] Differentiation of Implicit
Page 215 and 216:
Sec. 9]_Differentiation of Implicit
Page 217 and 218:
Sec. 10]_Change of Variables_2H 196
Page 219 and 220:
Sec. 10]_Change of Variables_213 or
Page 221 and 222:
Sec. 10] Change of Variables 215 an
Page 223 and 224:
Sec. 11]_The Tangent Plane and the
Page 225 and 226:
Sec. 11]_The Tangent Plane and the
Page 227 and 228:
Sec. 12] Taylor's Formula for a Fun
Page 229 and 230:
Sec. 13]_The Extremum of a Function
Page 231 and 232:
Sec. 13] The Extremutn of a Functio
Page 233 and 234:
Sec. 14] Finding the Greatest and S
Page 235 and 236:
Sec. 14] Finding the Greatest and S
Page 237 and 238:
Sec. 15] Singular Points of Plane C
Page 239 and 240:
Sec. 16] Envelope 233 Solution. The
Page 241 and 242:
Sec. 18] The Vector Function of a S
Page 243 and 244:
Sec. 18]_The Vector Function of a S
Page 245 and 246:
Sec. 19] The Natural Trihedron of a
Page 247 and 248:
Sec. 19] The Natural Trihedron of a
Page 249 and 250:
Sec. 20] Curvature and Torsion of a
Page 251 and 252:
Sec. 20]_Curvature and Torsion of a
Page 253 and 254:
Sec. 1] The Double Integral in Rect
Page 255 and 256:
Page 257 and 258:
Page 259 and 260:
Sec. 2]_Change of Variables in a Do
Page 261 and 262:
Sec. 2]_Change of Variables in a Do
Page 263 and 264:
Sec 3]_Computing Areas_257 2177. Co
Page 265 and 266:
Sec. 5]_Computing th Area* of Surfa
Page 267 and 268:
Sec. 6]_Applications of the Double
Page 269 and 270:
Sec. 7] Triple Integrals 2o3 Evalua
Page 271 and 272:
Sec 7\ Triple Integrals 265 We ther
Page 273 and 274:
gee. 7]_Triple Integrals_267 2251.
Page 275 and 276:
Sec. 8] Improper Integrals Dependen
Page 277 and 278:
Sec. 8]_Improper Integrals Dependen
Page 279 and 280:
See. 9] Line Integrals 273 Test for
Page 281 and 282:
Sec. 9]____Line Inlegtah____275 whe
Page 283 and 284:
Sec. 9]_Line Integrals__277 express
Page 285 and 286:
Sec. 9] Lim Integrals 279 (/-axis;
Page 287 and 288:
Sec 9] Line Integrals 281 Evaluate
Page 289 and 290: Sec. 9] Line Integrals 2S& D. Appli
Page 291 and 292: Sec. 10] Surface Integrals 285 When
Page 293 and 294: ec. 11]_The Ostrogradsky-Gauss Form
Page 295 and 296: Sec. 12] Fundamentals of Field Theo
Page 297 and 298: Sec. 12] Fundamentals of Field Theo
Page 299 and 300: Sec. 1. Number Series Chapter VIII
Page 301 and 302: Sec. 1]_Number Series_295 c) D'Alem
Page 303 and 304: Sec /\_Number Series_297 Note. For
Page 305 and 306: 2418. + + 0419 J___1_ . 4- 4- 4- '
Page 307 and 308: Sec. 1] Number Series 2464. cos-2-)
Page 309 and 310: Sec. 1]_Number Series_303 - V r(2-Q
Page 311 and 312: Sec. 2]_Functional Series_305 In th
Page 313 and 314: Sec. 2]_Functional Series_307 [the
Page 315 and 316: Sec. 2] Functional Series 309 Deter
Page 317 and 318: Sec. 3]_Taylor's Series_311 v8 s r
Page 319 and 320: Sec. 3] Taylor's Series 313 Since a
Page 321 and 322: Sec 3} Taylor's Series 315 2615. ln
Page 323 and 324: Sec. 3] Taylor's Series 317 2651. F
Page 325 and 326: Sec. 4] Fourier Series 319 If a fun
Page 327 and 328: Sec. 4\__Fourier Series _321 2690.
Page 329 and 330: Sec 1 1 Verifying Solutions 323 It
Page 331 and 332: Sec. 21 First-Order Differential Eq
Page 333 and 334: Sec. 3] Differential Equations with
Page 335 and 336: Sec. 3\ Differential Equations with
Page 337 and 338: Sec. 4]_First-Order Homogeneous Dif
Page 339: Sec. 5] Bernoulli's Equation 333 Co
Page 343 and 344: Sec. 7] First-Order Differential Eq
Page 345 and 346: Sec 8]_The Lagrange and Clairaut Eq
Page 347 and 348: Sec. 9] Miscellaneous Exercises on
Page 349 and 350: Sec. 9] Miscellaneous Exercises on
Page 351 and 352: Sec. 10] Higher-Order Differential
Page 353 and 354: Sec. 10]__Higher-Order Differential
Page 355 and 356: Sec. 11]_Linear Differential Equati
Page 357 and 358: Sec. 12] Linear Differential Equati
Page 363 and 364: Sec. 14]_Euler's Equations__357 Fin
Page 365 and 366: Sec. 15]_Systems of Differential Eq
Page 367 and 368: .Sec. 16} Integration of Differenti
Page 369 and 370: Sec. 17] Problems on Fourier's Meth
Page 371 and 372: Sec. 17]_Problems on Fourier's Meth
Page 373 and 374: Chapter X APPROXIMATE CALCULATIONS
Page 375 and 376: Sec. 1] Operations on Approximate N
Page 377 and 378: Sec. /]_Operations on Approximate N
Page 379 and 380: Sec, 2] Interpolation of Functions
Page 381 and 382: Sec. 2] Interpolation of Functions
Page 383 and 384: Sec. 3]_Computing the Real Roots of
Page 389 and 390: Sec. 4] Numerical Integration of Fu
Page 391 and 392:
Sec. 5] Numerical Integration of Or
Page 393 and 394:
Page 395 and 396:
Page 398 and 399:
Table 3. Basic Table for Calculatin
Page 400 and 401:
394 Approximate Calculations \Ch 10
Page 402 and 403:
ANSWERS Chapter I 1. Solution. Sinc
Page 404 and 405:
398 Answers Form a table of values:
Page 406 and 407:
400 Answer* 271. Solution. If x & k
Page 408 and 409:
402 Answers b) /'(!)= lim 1^15^= li
Page 410 and 411:
404_Answers 494. , . 495. - - . 1 2
Page 412 and 413:
-406 Answers 640. Hint. The equatio
Page 414 and 415:
408_Answers_ 755. e xcosa sin(*sina
Page 416 and 417:
410_Answers_4 where R is the radius
Page 418 and 419:
412 Answers 1/~JT f Q \ J/max = TT=
Page 420 and 421:
414 Answers 976. j/mi n =1.32 when
Page 422 and 423:
416 Answers 1037. j/ 1040. 3x* 13 J
Page 424 and 425:
418 1 1137. 4- 6 3a . In I ocot3x|
Page 426 and 427:
420 Answers 4^8 Problem 1218*. 1221
Page 428 and 429:
422 Answers X arctan(* + 2). 1294.
Page 430 and 431:
424 Answers sin 5* , 1374. -!- 1377
Page 432 and 433:
426 Answers 3 1442. In l-x2 ' '" 2
Page 434 and 435:
428_Answers_ the function /(x) = --
Page 436 and 437:
430 Answers 1651. 3na 2 . 1652. n(b
Page 438 and 439:
432 Answers inertia /=Y 2jttf 1 r *
Page 440 and 441:
434 Answers along the large lower s
Page 442 and 443:
436 Answers cos^. ,810. * ox x* x d
Page 444 and 445:
438_Answers 1905. 1914. u(*. !/) =
Page 446 and 447:
440 Answers 1983. 3* + 4j/-H2z 169
Page 448 and 449:
442 Answers is ~ |/2 + -~=, the alt
Page 450 and 451:
444 Answers (binormal). 2103. bx *=
Page 452 and 453:
444_Answers_x = \ ' > (principal no
Page 454 and 455:
446 Answers 2 a a a 2139. ( dy(f(x,
Page 456 and 457:
448_Answers_ r 4 **/* [ rj-cos*(p--
Page 458 and 459:
450 Answers 2243. 2244. 1 i VT=T* ^
Page 460 and 461:
452 Answers 2 t 2297. ) ~ll . |-[(I
Page 462 and 463:
454_Answers__ TtD the flux, use the
Page 464 and 465:
456__Answers_ and when x | ^ =- , t
Page 466 and 467:
458 Answers
Page 468 and 469:
460 Answers %-l n=o (n)=^=%. 2873.
Page 470 and 471:
462 Answers Chapter IX 2704. Yes. 2
Page 472 and 473:
464 Answers 2822. g-C*+; y=2*. 2824
Page 474 and 475:
466 Answers x + C z . 2914. #=
Page 476 and 477:
468_Answers_ A $007. 1) x = C cosco
Page 478 and 479:
470 Answers 3081. x = C l e 3082. x
Page 480 and 481:
472 Answers ,*. f (2"+ 1)JtJf dx. H
Page 482 and 483:
474 Answers 3131. a) 0.211; 0.389;
Page 484 and 485:
476 Appendix III. Inverse Quantitie
Page 486 and 487:
478 Appendix IV. Trigonometric Func
Page 488 and 489:
480 Appendix VI. Some Curves (for R
Page 490 and 491:
482 Appendix 11. Graphs //-=secx an
Page 492 and 493:
486 Appendix 24. Strophoid, * u s^-
Page 494 and 495:
Absolute error Absolute value 367 o
Page 496 and 497:
490 Index Differential equations of
Page 498 and 499:
492 Index Infinite discontinuities
Page 500 and 501:
494 Index Operator Hamiltonian 288
Page 502:
496 Index Taylor's series Term 311,
show all

Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

Create successful ePaper yourself

Delete template?

Save as template?