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

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Chapter IX DIFFERENTIAL EQUATIONS Sec. 1. Verifying Solutions. Forming Differential Equations of Families of Curves. Initial Conditions 1. Basic concepts. An equation of the type (n) F(x, = y, y' t/) 0, (1) where y y(x) is the sought-for function, is called a differential equation of order n. The function y = y(x), which converts equation (1) into an identity, is called the solution of the equation, while the graph of this function is called an integral curve. If the solution is represented implicitly, O(A, f/)~0, then it is usually called an integral Example 1. Check that the function t/ = sinjt is a solution of the equation Solution. We have: and, consequently, The integral /" + y sin x -f sin x ^ 0. */, C p .... C w ) = (2) of the differential equation (1), which contains n independent arbitrary constants C t , ..., C n and is equivalent (in the given region) to equation (1), is called the general integral of this equation (in the respective region). By assigning definite values to the constants C,, ..., C n in (2), we get particular integrals. Conversely, if we have a family of curves (2) and eliminate the parameters Cj C n from the system of equations = 0, -0. .... *-0, dx dx n we, generally speaking, get a differential equation of type (1) whose general integral in the corresponding region is the relation (2). Example 2. Find the differential equation of the family of parabolas y^C^x-CJ. (3) Solution. Differentiating equation (3) twice, we get: /' = 2C l (* C 2 ) and (/" = 2C,. (4) Eliminating the parameters C, and C 2 from equations (3) and (4), we obtain the desired differential equation
Sec 1 1 Verifying Solutions 323 It is easy to verify that tha function (3) converts this equation into an identity. 2. Initial conditions. If for the desired particular solution y = y(x) of a differential equation the initial conditions y (n) = f(x, y, y' y (n ~ l) ) (5) are given and we know the general solution of equation (5) y =
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TEXT FLY WITHIN THE BOOK ONLY
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OUP-43 30-1.71 5,000 OSMANIA UNIVER
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T. C. BapaneHKoe. C. M. Koean, r Jl
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TO THE READER MIR Publishers would
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Contents Sec. 6. Integrating Certai
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8 Chapter Contents Sec. 13. Linear
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Sec. 1. Functions Chapter I INTRODU
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Sec 1] Functions 13 8. Find the rat
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Sec. 1]_Functions_15 34. Prove that
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Sec. 2] Graphs of Elementary functi
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Sec. 2] Graphs of Elementary functi
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Sec. 2] Graphs of Elementary Functi
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Sec. 3] Limits 23 For the existence
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Sec. 3] 179. Hm (Vn + 1 \f~n). n -+
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Sec. #] Limits 27 209. lim 210. lim
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^Sec 3] Limits Therefore, Example 9
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Sec. 31 Limits 31 269. a) 270. lim-
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Sec. 4] Infinitely Small and Large
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Sec. 4] Infinitely Small and Large
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Sec. 5] Continuity of Functions 37
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Sec. 5]_Continuity of Functions_39
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Sec. 5] Continuity of Functions 334
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Sec. 1] Calculating Derivatives Dir
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Sec. 1] Calculating Derivatives Dir
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Sec. 2]__Tabular Differentiation_47
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Sec. 2]_Tabular Differentiation_49
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Sec. 2] Tabular Differentiation_ 51
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Sec. 2] Tabular Differentiation 504
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Sec. 2] Tabular Differentiation 55
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Sec. 3] The Derivatives of Function
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Sec. 3] The Derivatives of Function
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Sec 4] Geometrical and Mechanical A
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Sec. 4\ Geometrical and Mechanical
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Sec. 4] Geometrical and Mechanical
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Sec. 5]_Derivatives of Higher Order
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Sec. 5]_Derivatives of Higher Order
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Sec. 6] Differentials of First and
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Sec. 6}_Differentials of First and
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Sec. 7]_Mean-Value Theorems_75 745.
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Sec. 8] Taylor's Formula 77 765. a)
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Sec. 9]_ V Hospital-Bernoulli Rule
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Sec. 9] L' Hospital-Bernoulli Rule
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Chapter III THE EXTREMA OF A FUNCTI
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Sec. 1] The Extrema of a Function o
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Sec. 2]_The Direction of Concavity.
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Sec. 3]_Asymptotes_93 Example 2. Fi
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Sec. 3]_Asymptotes_95 Solution. Sin
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Sec 4] Graphing Functions by Charac
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CM CM -h CO O non ' > C " -~ a - >
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Sec. />] Differential of an Arc. Cu
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Sec 5] Differential of an Arc. Curv
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Sec. 5]_Differential of an Arc. Cur
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Sec. 1. Direct Integration 1. Basic
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Sec 1] Direct Integration_109 1048*
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Sec. 2] Integration by Substitution
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Sec. 2] Integration by Substitution
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Sec. 3\_Integration by Parts_H7 Exa
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Sec 4] Standard Integrals Containin
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Sec. 5] Integration of Rational Fun
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Sec. 5] In t egration of Rational F
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Sec. 6] Integrating Certain Irratio
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Sec. 6] Integrating Certain Irratio
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Sec. 7]_Integrating Trigonometric F
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Sec. 7]_Integrating Trigonometric F
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Sec. 8] Integration of Hyperbolic F
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Sec. 11] Using Reduction Formulas 1
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Sec. 12] Miscellaneous Examples on
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Sec. 1] The Definite Integral as th
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Sec. 2]_Evaluating Definite Integra
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Sec. 3]_Improper Integrals_143 Sec.
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Sec. 3] Improper Integrals 145 Solu
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Sec. 4] Change of Variable in a Def
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Sec. 5] Integration by Parts 149 Bu
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Sec. 6] Mean-Value Theorem 151 In p
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Sec. 7] The Areas of Plane Figures
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Sec. 7] The Areas of Plane Figures
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Sec. 7\_The Areas of Plane Figures_
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Sec. 8] The Arc Length of a Curve 1
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Sec, 9] Volumes of Solids 161 1673.
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Sec. 9] Volumes of Solids 163' Solu
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Sec. 9]_Volumes of Solids_165 1693.
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Sec. 10] The Area of a Surface of R
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Sec. II] Moments. Centres of Gravit
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Sec 11] Moments. Centres of Gravity
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&?c. 12] Applying Definite Integral
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Sec. 12] Applying Definite Integral
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Sec. 1) Basic Notions [SI Putting *
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Sec. 1]_Basic Notions_183 1790*. Le
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Sec. 3] Partial Derivatives 185 180
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Sec. 4]_Total Differential of a Fun
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Sec. 4}_Total Differential of a Fun
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Sec. 5]_Differentiation of Composit
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Sec. 6]_Derivative in a Given Direc
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Sec. 6] Derivative in a Given Direc
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Sec. 7]_Higher-Order Derivatives an
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Sec. 7] Higher-Order Derivatives an
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Sec. 7]_Higher-Order Derivatives an
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Sec. ti\_Integration of Total Diffe
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Sec. 9]_Differentiation of Implicit
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Sec. 9] Differentiation of Implicit
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Sec. 9]_Differentiation of Implicit
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Sec. 10]_Change of Variables_2H 196
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Sec. 10]_Change of Variables_213 or
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Sec. 10] Change of Variables 215 an
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Sec. 11]_The Tangent Plane and the
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Sec. 11]_The Tangent Plane and the
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Sec. 12] Taylor's Formula for a Fun
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Sec. 13]_The Extremum of a Function
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Sec. 13] The Extremutn of a Functio
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Sec. 14] Finding the Greatest and S
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Sec. 14] Finding the Greatest and S
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Sec. 15] Singular Points of Plane C
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Sec. 16] Envelope 233 Solution. The
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Sec. 18] The Vector Function of a S
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Sec. 18]_The Vector Function of a S
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Sec. 19] The Natural Trihedron of a
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Sec. 19] The Natural Trihedron of a
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Sec. 20] Curvature and Torsion of a
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Sec. 20]_Curvature and Torsion of a
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Sec. 1] The Double Integral in Rect
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Sec. 2]_Change of Variables in a Do
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Sec. 2]_Change of Variables in a Do
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Sec 3]_Computing Areas_257 2177. Co
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Sec. 5]_Computing th Area* of Surfa
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Sec. 6]_Applications of the Double
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Sec. 7] Triple Integrals 2o3 Evalua
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Sec 7\ Triple Integrals 265 We ther
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gee. 7]_Triple Integrals_267 2251.
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Sec. 8] Improper Integrals Dependen
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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: Sec. 4\__Fourier Series _321 2690.
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 and 340: Sec. 5] Bernoulli's Equation 333 Co
Page 341 and 342: Sec. 6] Exact Differential Equation
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
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Sec, 2] Interpolation of Functions
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Sec. 2] Interpolation of Functions
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Sec. 3]_Computing the Real Roots of
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Sec. 4] Numerical Integration of Fu
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Sec. 5] Numerical Integration of Or
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Table 3. Basic Table for Calculatin
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394 Approximate Calculations \Ch 10
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ANSWERS Chapter I 1. Solution. Sinc
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398 Answers Form a table of values:
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400 Answer* 271. Solution. If x & k
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402 Answers b) /'(!)= lim 1^15^= li
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404_Answers 494. , . 495. - - . 1 2
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-406 Answers 640. Hint. The equatio
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408_Answers_ 755. e xcosa sin(*sina
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410_Answers_4 where R is the radius
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412 Answers 1/~JT f Q \ J/max = TT=
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414 Answers 976. j/mi n =1.32 when
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416 Answers 1037. j/ 1040. 3x* 13 J
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418 1 1137. 4- 6 3a . In I ocot3x|
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420 Answers 4^8 Problem 1218*. 1221
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422 Answers X arctan(* + 2). 1294.
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424 Answers sin 5* , 1374. -!- 1377
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426 Answers 3 1442. In l-x2 ' '" 2
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428_Answers_ the function /(x) = --
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430 Answers 1651. 3na 2 . 1652. n(b
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432 Answers inertia /=Y 2jttf 1 r *
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434 Answers along the large lower s
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436 Answers cos^. ,810. * ox x* x d
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438_Answers 1905. 1914. u(*. !/) =
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440 Answers 1983. 3* + 4j/-H2z 169
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442 Answers is ~ |/2 + -~=, the alt
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444 Answers (binormal). 2103. bx *=
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444_Answers_x = \ ' > (principal no
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446 Answers 2 a a a 2139. ( dy(f(x,
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448_Answers_ r 4 **/* [ rj-cos*(p--
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450 Answers 2243. 2244. 1 i VT=T* ^
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452 Answers 2 t 2297. ) ~ll . |-[(I
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454_Answers__ TtD the flux, use the
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456__Answers_ and when x | ^ =- , t
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458 Answers
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460 Answers %-l n=o (n)=^=%. 2873.
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462 Answers Chapter IX 2704. Yes. 2
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464 Answers 2822. g-C*+; y=2*. 2824
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466 Answers x + C z . 2914. #=
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468_Answers_ A $007. 1) x = C cosco
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470 Answers 3081. x = C l e 3082. x
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472 Answers ,*. f (2"+ 1)JtJf dx. H
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474 Answers 3131. a) 0.211; 0.389;
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476 Appendix III. Inverse Quantitie
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478 Appendix IV. Trigonometric Func
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480 Appendix VI. Some Curves (for R
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482 Appendix 11. Graphs //-=secx an
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486 Appendix 24. Strophoid, * u s^-
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Absolute error Absolute value 367 o
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490 Index Differential equations of
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492 Index Infinite discontinuities
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494 Index Operator Hamiltonian 288
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496 Index Taylor's series Term 311,
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