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

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464 Answers 2822. g-C*+; y=2*. 2824. L=4 2823 l* = nlPL-zIl sin P + C '2825.< , Hint. The differential ' equation from which x is defined as a function of p is homogeneous. 2826. = . 2827. j/ = Cx + C; no singular solution 2828. * + y*=\. 2829. = 2 */ Cx + = ; / 4*. 2830. -g JO/-C 2831. A circle and the family of its tangents. 2832. The astroid x*" + y*i> = a z i>. 2833. a) Homogeneous, y=xu\ b) linear in x\ x = uv\ c) linear in y\ y = uv, d) Bernoulli's equation; y = uv\_e) with variables separable; f) Clairaut's equation; reduce to y^xy' V^y^l g) Lagrange's equation; differentiate with respect to x\ h) Bernoulli's equation; y = uv\ i) leads to equation with variable? separable; u--x-}-y\ j) Lagrange's equation; differentiate with respect to x\ k) Bernoulli's equation in r, x=uv\ 1) exact differential equation; m) linear; y = uv\ n) Bernoulli's equation; y = uv. 2834. a) sin-^ = In | jc| + C; b) x^y- 2835. x 2 4 + = 1 . (/ Q/ 2836. /=i~. 2837. xi/(C-~ In 1 x)= 1. 2838. lnC; singular solution, i/ = f- (x+l) . 2839. y=-Cx+ VaC\ singular solution, y = ~. 2840. 3// + ln L~1J = C. 2841. ***-^ -arc tan y~ . 2842. y = x(l+Ce*). 2843. Ar= 1 ). - X -f- 1 2844. y = 2847. x = Ce sIn ^-2a(l+sini/). 2848. i + Sx + y + ln [(x-3) 10 2849. 2 arc tan -= In Cx. 2850. **=! --- \-Ce 2 1 y- 1 | 8 ] = C. y . 2851. x 3 =-Ce^ y 2 2852. l- + ln|x| = C. 2853. (/ = * arc sin (Cx). 2854. y i ^Ce" ix + sin jc+ r X 5 + 4- cos*. 2855. xy = C(y 1). 2856. x = C^-- (sin y + cos /). 2857. pyo ^ 2860. 2863. y = -l. 2858. x = CeV-
-0. 2873. y^Cx + -~, i/ = ~^/2?. 2874. x'-f-x 2 */ y 2 9 x ^C. 2875. ij p 2 + 4# 2 ^Q/ 3 . 2876. 0=-=* 1. 2877. r/-=x. 2878. = / 2. 2879. f/-0. 2880. i/^-^- (sin x + cos x). 2881. y = -i- f (2x + 2x + l). 2882. y = ---=e-* + 2x 2. 2883. a) y = x; b) = t/ Cx, where C is arbitrary; the point (0,0) is a singular point of the differential 2 equation. 2884. a) y = 2 x\ h) # = 2px; (0,0) is a 2 singular point. 2885. a) (x C) 4-(/ 2 -=C 2 ; b) no solution; c) x 2 2 + t/ -=x; (0,0) is a singular point. 2886. // --e". 2887. y^(V~2a Kx) 2 2 . 2888. ^ f/ --1 e~ x . 2889. r-=Ce^. Hint. Pass to polar coordinates. 2890. 3y 2 2-=0 2891. 2892. x 2 -|- (// 6) 2 -_6 2 2 . 2893. // [- 16x^0. 2894. Hyperbola 8 i/ r-^/fq) .v f --C or circle x 2 i-j/ 2 -C 2 . 2895. t/--i- (e- v + e- x ). Hint. Use tha fact A: :c that the area is equal to \ J y dx and the arc length, to f / | 2 H^l/ -^^ o 2 2896. Cf/. 2897. - // 4C (C -j- a x). 2898. Hint. Use the fact that tha x=^^--f resultant of the force of s>ravit\ and the centrifugal force is normal to the surface. 'lakmg the r/-axih as the nxis of rotation and denoting by co the angular velocity "of rotation, \\ e get for the plane axial cross-section of the desired surface the differential equation g-'-'==o) 2 x. 2899. p_^ e --*-j--a; 2902. T^a + (TQ a)e~ ki . 2903. In / 3 V ~ one hour. 2904. (o^lOO(-g-J rpm. 2905. 4 2?6 of the initial quantity Q will decay in 100 years. Hint. Equation ^ =kQ. Q-=Q n^- 2906. t =^= (4-) =^35.2 sec. Hint. Equation n(/i f 2fc)d/i = n ~V udf. 2907. . Hint, as ' "^ ^ is a dQ== kQ dfi. Q = Qo (y)"* 2908int. Equation = mgkv m^t : g ;v= ~tanh ft y I/ . ~) 2909. 18.1 kg. Hint. Equation ~ =*f~V ~ 2910. / =__ [(/? sin wf 16-1900
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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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Sec. 8]_Improper Integrals Dependen
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See. 9] Line Integrals 273 Test for
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Sec. 9]____Line Inlegtah____275 whe
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Sec. 9]_Line Integrals__277 express
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Sec. 9] Lim Integrals 279 (/-axis;
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Sec 9] Line Integrals 281 Evaluate
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Sec. 9] Line Integrals 2S& D. Appli
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Sec. 10] Surface Integrals 285 When
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ec. 11]_The Ostrogradsky-Gauss Form
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Sec. 12] Fundamentals of Field Theo
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Sec. 12] Fundamentals of Field Theo
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Sec. 1. Number Series Chapter VIII
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Sec. 1]_Number Series_295 c) D'Alem
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Sec /\_Number Series_297 Note. For
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2418. + + 0419 J___1_ . 4- 4- 4- '
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Sec. 1] Number Series 2464. cos-2-)
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Sec. 1]_Number Series_303 - V r(2-Q
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Sec. 2]_Functional Series_305 In th
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Sec. 2]_Functional Series_307 [the
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Sec. 2] Functional Series 309 Deter
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Sec. 3]_Taylor's Series_311 v8 s r
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Sec. 3] Taylor's Series 313 Since a
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Sec 3} Taylor's Series 315 2615. ln
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Sec. 3] Taylor's Series 317 2651. F
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Sec. 4] Fourier Series 319 If a fun
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Sec. 4\__Fourier Series _321 2690.
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Sec 1 1 Verifying Solutions 323 It
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Sec. 21 First-Order Differential Eq
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Sec. 3] Differential Equations with
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Sec. 3\ Differential Equations with
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Sec. 4]_First-Order Homogeneous Dif
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Sec. 5] Bernoulli's Equation 333 Co
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Sec. 6] Exact Differential Equation
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Sec. 7] First-Order Differential Eq
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Sec 8]_The Lagrange and Clairaut Eq
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Sec. 9] Miscellaneous Exercises on
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Sec. 9] Miscellaneous Exercises on
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Sec. 10] Higher-Order Differential
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Sec. 10]__Higher-Order Differential
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Sec. 11]_Linear Differential Equati
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Sec. 12] Linear Differential Equati
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Sec. 14]_Euler's Equations__357 Fin
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Sec. 15]_Systems of Differential Eq
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.Sec. 16} Integration of Differenti
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Sec. 17] Problems on Fourier's Meth
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Sec. 17]_Problems on Fourier's Meth
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Chapter X APPROXIMATE CALCULATIONS
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Sec. 1] Operations on Approximate N
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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
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 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,
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