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S I. dy fx x fx y fx + C 3 C vt dy fx 4 x, y dy yt gt + Ct + C dt v e kt xt v e kt + C k x v k + C C xt v k 3 r r + dr e kt S Sr πr dt d v } dt k e kt k v k e kt v e kt [ ] S ds πrdr x± xt k < v k Sr ds πrdr πr + C lim t ± [ ] dv dt g dy dt vt vt gt + C yt v C xt y C v e kt k > xt yt gt [ ] v dt S C S πr g vt yt
S I 3 r r + dr N V V r [ ] Nt 4πr dn dt x dn dt kn x x + U Ux dn kdt 3 λ N x x + l log N kt + C N C m exp[ kt] 4 σ N N C N r r + dr N N exp[ kt ] k log T a M I [ Ma Nt N exp log ] T t.. dy fx gy 5 x fx y gy Ix gydy fx 6 5 y x m v Stokes gydy fx + C 7 5 ambert x m dv mg kv dt k dv mg k v k m dt mg 4 log k v k m t + C v C T exp km t + mg k [ ] Newton m v f
S I 4 v 6 vt mg k exp km t } v vt v mg k [ ] G M Newton η..8.6.4. /e -exp-.5*x 4 6 8 4 7 v v yt 8 v v vt 8 v g 6 v v 4 9 Newton T T t T w r v T a k 6πηa dv ρ m 4πρa 3 /3 dt MG r g r v mg/k ρga /9η a g MG dv dt dr dv dt dr v dv dr v dv dr g r v g + C r r v v g g r + v 6 v v vt v ± g r + v v v r v r v k
S I 5 g > v v > g v > g v rt v rt 7 l λ r u ρ l du u x g u u dt xt u v ξ g g i ξ > v < v g v T gξ t [ ρ + tan ρξ ξ ρξ + ] tan ξ ξ ξ F ρ gξ t gξ T F ρ ii ξ g t 3 3 ρ 3 t [sec] e+8 e+7 e+6 e+5 e+4 e+3.9.99.8.7.9999 cx 3/ e+ e+ e+ e+ e+3 e+4 r/ r t x x l [ ] xt vx vx xt λ l dv dt λl + xg λl xg λgx dv dt v dv vx v dv g l x. v, x v g l x + C t x x, v C g dt v l x x l g log x + x x t + C x C l g log x + x x t x
S I 6 x g e l t + e g l t g x x x cosh l t hyperbolic [ ] C A B g l cosh t sinh t + d d cosh t sinh t, dt dt x sinh t sinh t cosh t dt v x x x cosh t x x cosh t x sinh t xt x C a/b x v dv dv dt x cosh t g l x T T.8 [ ] x.6 x T x λ l g l.4 λ l. x xt 3 4 5 T x [ ] xt 3 C a., b.9 T x T x λ x xg l a > b µ > 3 r x b, ẋ 8 A + B C C x A B a b C k dt ka xb x kdt a xb x a x b x kt + C a x a b log b x a x b x C e µt µ a b k x ab eµt e µt ae µt be µt. b a a b 4 A X A X X
S I 7 xt A a X S x A X a A X k, k yt y H [ ] 9 x dt t dv avdt v x s kx Torricelli v gy dt k [ ] dv a gy dt dt s kxx dt s kxx x + k s kx s t + C } log x logs kx s s log x s kx C e st x s kx x se st C + ke st < x /.5 - C e- C e s C + k < s k C dv dv Sdy dv dv a gydt Sdy S S y a g a.5 yt ga S S a S y t + C a g y H C H g t H g t t t C e C e -6-4 - 4 6..8.6 4 k, s -t.4 A B A B..5.5 5
S I 8 5 n C a yt k + n k n ekαt n kekαt I e kαt + α tanh Iαt yt tanhx tanh± ± y y ytanhx 6 y ry yt t ry x 7 w H y - ht g [ ] ht t + H 3π w gh h rt 3 n 9 Stephan-Boltzmann αn T T α e 4 Boltzmann k I Mc dt dt kst 4 Te 4 dn dt I αn nt t T t [ ] [ ] T t T e k I/α αdt dn k n k k + n + Boyle Mariotte k n V p V/p log k + n k n kαt + C k + n k n C e kαt 6 tanhx 8 cm.5cm M c S
S I 9 Te.8Te.6Te.4Te.Te.5.5.5 3 3.5 4.3 dy y f x i σe P l i + ρ/ t E O ρ/ε v v B O σ 7 Ω m [ ] O OP x y θ dt v dy B cos θ, dt v v B sin θ dy v B sin θ v v B cos θ i log x + a x + a r x + y, x r cos θ, y r sin θ ii x + a a arctan x a dy a iii x a a log x a y β x + y β v x v B x + a a iv a x arcsin x a arccos x y ux a > a x v x + a log u + x du u β + u + x + a vi du ax + bx + c log x b 4ac log ax + b β + u β log u + + u + C b 4ac ax + b + x l y u b 4ac C log l b 4ac> β log x log u + + u arctan ax + b l 4ac b 4ac b 8 u y x x, u log x du fu u + C 9 u y x y x Px, y OP x ax + b b 4ac< b 4ac x l β u + + u x } β u + u l u x β x } β l l
S I.5 y l x β x } +β l l θ r l c l l θ r tan + π β cos θ 4 x r cos θ β l θ tan x + π 4 β l tan θ x + tan θ + tan θ tan θ tan θ β } β l l x x y x tan θ.5..4.6.8 t x v v B l 7 v B > v β < O v v B.5 O v B < v l β > x β x y dr dt v B + v sin θ r dθ dt v r cos θ.5 dr dr rdθ dt β.8 r dθ β.5 dt rθ β. dr r vb v cos θ tan θ..4.6.8 log r v B θ log tan v + π log cos θ + log c 4 8
S I [ ] t x t l x β+ v B β l β x } +β + β l.4 x px, qx y [ y ] dy + pxy qx y β β l qx 3 yx y ce px cwx ii 6yy + 9x iii y + y iv y + x + y v x + yy λy vi xy x + y vii xy y x 3 + y viii y y + x y x c x ux 4 u du du wx qx qx wx qx 5 u + c wx i y xy agrange qx } y e px qx e px + C + 3 A λa B λb C A, B, C N A,, B n B t [ ] B A λ A C λ B dn B λ A n A t λ B n B t dt dn B + λ B n B t λ A n A t dt n A t n A N A n A dt λ an A n A t N A e λ At e ± px e ± λ B dt e ±λ Bt λ A
S I n B t e λ Bt } λ A N A e λat e λbt dt + C } e λ λa N Bt A e λ B λ A t + C λ B λ A n B C.7.6.5.4.3.. n B t e λ λ Bt A N } A e λ B λ A t λ B λ A λ AN A λ B λ A e λ A t e λ Bt } e λ Bt I i Et E.8 λ λ.99 λ.5 4 6 8 λ. λ. 9 A B C B -.5 6 g v i Et E } It e t E e t + C ii Et E sin t } It e t E e t sin t dt + C Ce t + E + It E Ce t + E sin t cos t e t ii Et E sin t } E It e + t + sin t cos t -....3.4.5..5 4 H, Ω, Hz, E 5 V Et It Et E Et E sin t [ ] di di + I Et dt dt + I Et Et It e t } Et e t dt + C, Et
S I 3.5 ON Θt t < t > F x, y, z x +y c Dirac. δx x,. δx afx fa δx y x y aplace 7 C C Et It Et E Et E sin t 8 C C It F x, y, c Q y fx, y 3 dy + pxy qxyn c F x, y, c xy c c parameter c c c 9 y 4 fx, y Bernoulli v y n 5 xy q dv n pxv n qx [ ] V r vx q 4πε r xy q c 4πε x + y c q x + yy y x fx, y 4πε x + y 3 y
S I 4 y fx, y y x y cx 3 λ P,, Q, y + x + } 4x c 3 λ 3 +λ, λ P,, Q, x + c + y c 3
S I 5 y yt [ y ] s [ y ] y s [ y ] sy y i [ y i ] s i [ y ] s i j y j 5 aplace [ n ] aplace a i y n i [ f ] i aplace n n i a i s i [ y ] a i s i j y j i i j. aplace y n + a y n + + a n y + a n y fx 5 s aplace y [ [ y ] ] j [ y ] [ f ] + n i a i i s i j y j j n a i s i i.... fx s F s e st ft dt 6 ft aplace [ y ] y [ ] e st y dt e st y +s e st y dt 7 lim t e st yt s > 8 yt 7 e st y dt y + s e st y dt 9 [ y ] s [ y ] y 6 ft [ ] [ ] s e st dt s [ ] e st 7 ft e at [ ] [ e at ] e st e at dt [ ] e s at s a s a
S I 6. aplace 33 z a + i [ e zt ] s z s y 34 [ cosh t ] [ sinh t ] 7 8 ft aplace I F s s s > t n n : n! s n+ s > 3 t x x > Γ x + s x+ x > 4 e at s > a s a s 5 cos t s + s > 6 sin t 7 cosh t 8 sinh t s + s > s s s > s s > 9 t n e at n! s a n+ s > a e at cos t e at sin t t cos t 3 t sin t s a s a + s > a s a + s > a s s + s s +.. 9 y ky [ ] s [ y ] y + k [ y ] [ y ] y s + k s + k e kt [ ] y y ye kt s + k 35 y ky + b 36 y + ky e t y + y [ ] s [ y ] sy y + [ y ] 8 4 a i [ y ] sy + y [ cos t ] [ cos t ] s + [ ] 4 a i s + sin t [ e it ] s i s + i s s + s + cos t s s + + i s + yt y cos t + y sin t aplace [ e it ] [ cos t + i sin t ] [ cos t ] + i [ sin t ] 5,6 37 y y 38 y + y cosωt
S I 7.. 3 Γ p y + P y + Qy ft P Q 3 ft Γ p u p e u u p > ft 3 aplace Γ [ ] aplace [ y ] sy + P y + y s + sp + Q gs s pγ p + sp + Q 4 p n i gs gs s β s β Γ n + nγ n nn Γ n [ y ] sy + P y + y gs A s β + A s β A β y + P y + y β β A β y + P y + y β β yt A e β t + A e β t Γ p + β.3 P, Q ii gs s + P s Q gs ss + P e u du u p e u du [ ] u p e u + p nn Γ n! u p e u du ft t x [ t x ] e st t x dt st u u x s x e u s du Γ x + s x+ s x+ u x e u du [ y ] sy + P y + y gs B s + P + B s B y P B y + y P.3. f t f t α, β [ αf t + βf t ] α [ f t ] + β [ f t ] 5 yt y P e P t + y + y P [ ] [ ] P [ αf t + βf t ]
S I 8 α e st αf t + βf t } dt e st f t dt + β e st f t dt.3.4 t ft ft-a α [ f t ] + β [ f t ] a Θ t-a Θ t-a ft-a.3. a a ft F s [ f ] s s 4 s F s s s a ft [ ft e at t ] f d s [ ft ] ft t t a F s e as t t a < t a [ ] ft gt f d ft Θ a t [ ft ] [ g t ] s [ gt ] g t < a Θ a t Θt a t > a g... ft [ f ] s [ g ] t ft.3.3 s ft F s e at ft 3 t [ e at ft ] e st e at ft dt [ ] F s a e s at ft dt t [ Θt aft a ] e as F s 7 [ Θt aft a ] F s a s s a t a s F s a e sa e s f d e sa F s s F s [ e at ft ] F s a 6 4 Θ a t [ ] ft s t 39 9 s [ Θ a t ] [ Θ a t ] e sa [ ] e sa s a e st Θt aft a dt e st a sa ft adt
S I 9 5 t a E [ ] It Y [ y ] [ ] Et Θ a te di dt + I Θ ate di dt + I Θ at E Y e s ss + s + 3 e s F s s [ I ] I + [ I ] E e as s I e as ft [ F s ] 3 e t + 6 e 3t [ I ] E s + s E e as s s + E } e as e as s s + It E } s s + e t t yt [ Y ] ft ft Θ t } e t a Θ a t } It E e t t a 4 It s Y + 4sY + 3Y s e s Y /3 F s s / s + + /6 } s + 3 t [ e s F s ] ft Θ t 3 e t + 6 e 3t t < e e t 6 e3 e 3t < t 5.. 3 5 6 y + 4y + 3y gt, y, y t > p gt Θ t Θ t.4 ft + p ft t
S I p ft k e as s e as + e as k e as s + e as [ f ] e st k as tanh ft dt s p e st f dt + p p e st f dt + 3p p e st f dt + t + p 3 t + p n t + n p p ft f + p f, f + p f, [ f ] p e st f d + p + p e st f d + + e sp + e sp + e st f d p e st f d iii sin t ft e ps k + e sπ k t ft p s + e sπ/ π/ π/ 3π/ 4π/ p [ f ] e ps e st ft dt 8 8 7 vt V k -k ft a a 3a t 4 i k a 4a t ii sin t k ft π/ π/ 3π/ 4π/ -V t k as tanh as k s + e sπ/ t vt [ ] vt it + di dt V s Is + sis i} 6 [ ] p a 8 [ f ] e as e as e as a e st ft dt a a } ke st dt + ke st dt a k s e as + k } s e as e as Is i s + + V s s +, Is i t t ie t t V s V e T s s + e T s V s s +
S I V V V V e T s ss + + e T s } s e T s s + e T s } s s + e T s e T s n s s + n } n e nt s e n n+t s} gs.5 s s + ht e t t gse nt s ht t N NT t N + T.5 Θt nt i t V N n ht nt ht n+t } n + V N ht NT N V e T n e t T e n V e t T e T N e e T V e T e t e T N e t NT } + ξ t NT ξ T i ξ V e T e ξ + NT e T + e T e ξ i s ξ N V [ e ξ e T e T + e ξ N V [ e T e T + e ξ ] ] 7 T.,. T.,. nt Θt nt i t V -.5 n ht nt Θt nt n ht n+t Θt n+t } - 3 4 5 -.5-7.. 3 4 5 T.., i s t t T MT T M + T T T N V e T + i + N V } t V M n ht nt ht n+t } e ξ n + V [ ht MT M ht M +T Θt M +T ] i s ξ V e T e t e T e t MT } +
S I ξ t MT ξ T i s ξ V [ e T e T + e ξ e } ξ T Θξ T + e ξ V [ T ] e e T + e ξ [ T ] V e e T + e ξ T ] ξ < T T ξ < T 4 ii s + it V it + + sint nπ + β n [ exp t n π ] } Θ t n π + sin β, + cos β.8.6.4. -..5.4.3.. -. -.4...3.4.5.6 -. aplace V s V s Is s + s + e πs V s e πs s + s } s + V e πs + s e t n π e t 8..5 V n n e t e t M+π e π 4 C vt it + it dt di C dt + i C + dv it V dt aplace v i + sint + β e sv s t e M+π t [ i ] Is s + C e π V s + e πs s s + + V C e V C t πs + s + V e πs + s + V + s + + s s + + + s + + s s + it V + + cos t n π n s + + } s s + } s s + } s s + } n sin t n π e t n π } Θ e πs n t n π M π t < M + π M n M M sint nπ + β sint + β n cos nπ sint + β + + M e n π Θ t M + π }
S I 3 + e ξ t M π M+π t Θ t M + π } + iξ V + sinξ + β Θ ξ π } e ξ e Mπ e ξ π e π + e ξ π Θ ξ π } + i s ξ V + sinξ + β Θ ξ π } e ξ π ξ e π + Θ π } + e ξ sinξ + β e ξ e π e π e π e π ξ < π π ξ < π ii i i ξ + fξ cosξ γ e π ξ e π e ξ < π fξ ξ ξ v c ξ v c ξ fξ i ii V V π π π M 9.. V.8.6.4. -..5.4.3.. -.4...3.4.5.6 -. -. ξ v c ξ vξ i s ξ V sinξ Θ ξ π } i s ξ i ξ < π.8.6.4.5.4.3 ii V V sinξ sinξ + β + + V + V + sinξ γ + + sin γ cos γ +, π ξ < π V + e π e π e e π e π e π e π e e + ξ ξ ξ }. -. -.4.. -. -.6...3.4.5.6 -... V Fourier
S I 4 3 fx gx T hx + T αfx + T + βgx + T αfx + βgx hx hx T 3. fx x sinx cosx fx T fx x 9 T fx T fx a + a cos π T x + b sin π T x T + a cos 4π T x + b sin 4π T x + a k cos kπ T x + a k sin kπ T x 3 n 9 fx + nt fx T x 3 T 3T 4T fx fx gx T 45 n,,, 3, hx αfx + βgx 3 T 9 3 3 [ ] n fx + nt fx + n T + T fx + n T fx + n T fx 43 fx x T fax a T/a k k a n b n T T/k T/k T 44 π i iv vi / i fx ii fx iii fx x < x < < x < π cos x < x < < x < π π + x < x < π x < x < π iv fx x < x < π sin nx / ii x sin nx e x sin nx v vii cos nx / / iii x cos nx e x cos nx x sin nx
S I 5 3. [ ] a + a k cos kx + b k sin kx cos jx 37 T π k a sin jx + [a k cos kx sin jx k ] fx a + a k cos kx + b k sin kx 33 b k sin kx sin jx k b j π fx a k b k a 33 π fx [ ] a + a k cos kx + b k sin kx a + k k a k cos kx + b k sin kx πa + c b a n fx sin nx π π fx 34 π 39 47 46 k π π k j cos kx sin kx cos kx cos jx πδ kj π k j δ kj k, j a b j k j 33 cos jx π fx cos jx [ ] a + a k cos kx + b k sin kx cos jx 35 k a cos jx + [a k cos kx cos jx k ] b k sin kx cos jx 3 a j π k < x < fx + π fx, fx a j k < x < π fx cos jx j,, 36 π [ ] b, b, 33 sin jx π a fx [ ] k + k π π π fx sin jx [ kπ + kπ] π b j π fx sin jx j,, 38 j n a a π b a n π sin kx sin jx fx fx cos nx n,, cos kx cos jx πδ kj T π fx a, a, b, fx fx
S I 6 a n fx cos nx π [ ] k cos nx + k cos nx π π sin nx k sin nx π n + k n b n fx sin nx π [ ] k sin nx + k sin nx π π cos nx k cos nx π n k n 4k nπ n, 3, 5, 4k 3 5 sin x + sin 3x + sin 5x + π fx x x fx fx fx lim h fx h fx + fx + lim h fx + h x fx h fx lim h h fx + h fx + lim h h x π π f k 4k 3 5 7 + + π 3 + 5 7 + π 4 π 3.3 fx 48 π fx fx π x π i fx x < x < π ii fx x < x < π fx < x < iii fx fx x x < x < π x < x < iv fx π /4 < x < π 49 fx a n, b n kfx ka n, kb n fx x k
S I 7 3.4 3 ft sin t p. y + λy + y sin t E sin t p.7 3 y + P y + Qy ft P, Q ft e-4 e-5 3 Et It Et E sin t e-6 [ ] It di dt + I Et 3 It A cos t + B sin t.. di dt A sin t + B cos t A B + B cos t + A sin t E sin t t sin t, cos t A + B, B A E 4 A, B A E B E + + [ ] It yt A cos t + B sin t E It / + sin t cos t y t A sin t + B cos t 5 Et It Et E cos t y t A cos t + B sin t A + λb + A cos t + B λa + B sin t sin t
S I 8 sin, cos A, B 3.5 A + λb λa + B y + P y + Qy ft yt λ cos t + sin t + λ sint δ tan δ λ ft a + an cos n t + b n sin n t + λ n n D δ 4 π sin δ 33 π 5 C C Et sin t y + λy + y ft It < t < ft < t < π 5 C [ ] ft fourier Et sin t It ft 4 sin nt π n 5 C y + P y + Qy a n cos n t + b n sin n t D + λ y n λ δ arctan yt y t + y t + 3 n n:odd y + λy + y 4 sin nt nπ n : 4 y n t yt y t + y 3 t + y 5 t + 4 y n t a n cos nt + b n sin nt y nt nb n cos nt na n sin nt y nt n a n cos nt n b n sin nt
S I 9 4 3 n a n + λnb n + a n cos nt n b n λna n + b n sin nt 4 sin nt nπ sin, cos a n, b n n a n + nλb n nλa n + n b n 4 nπ a n 4 nπ nλ π n + nλ b n n nλ y n t 4 nλ cos nt + n sin nt nπ n + nλ 4 nπ sinnt δ n tan δ n + nλ nλ a n n D n δ n 4 D n nπ n + nλ nλ δ n arctan n 6 3., λ.5 [ ] p. D n n 3 3. n 3 7 n,3,5,7... 3 4 5 6 7 8 9 - - -3 6 4 i n t a n cos nt + b n sin nt π T 7 3 4 5 6 7 8 9 3 4 5 34 T it V -V vt t vt it dit + vt it dt vt vt 4V sin πt π T + 3πt sin 3 T + 5πt sin 5 T + dit + dt it 4V nπt sin n 4 nπ T i n t yt i t + i 3 t + i 5 t +
S I 3 di n t dt nb n cos nt na n sin nt 4 nb n + a n cos nt + na n + b n sin nt 4V sin nt nπ nb n + a n, na n + b n 4V nπ a n T 4V n + } b n n a n a n, b n 4V sin nt cos nt n i n t } T n + 4V πn n + sinnt δ n tan δ n n, n : 3 8.5 -.5-8 3 4 5 T,,., V 3
S I 3 3.6 > fx [ cos x fξ cos ξ dξ π ] Fourie sin x fξ sin ξ dξ d aplace fx [ ] a cos x+b sin x d π a fx cos x 43 b fx sin x 3.6. T f T x a + a n cos n x + b n sin n x a T a n T n T/ T/ T/ T/ T/ f T x f T x cos n x 4 b n f T x sin n x fx T T/ n nπ T fx 43 fx x n+ n n n + π T nπ T π T 35 f T x T/ f T ξdξ T T/ x < + [ T/ fx x > cos n x f T ξ cos n ξ dξ π T/ T/ sin n x f T ξ sin n ξ dξ T/ a T fx fx lim f T x T d b gn gd ] 3.6. fx [ ] a b fξ cos ξ dξ cos ξ dξ fξ sin ξ dξ sin ξ dξ sin ξ cos ξ sin
S I 3 fx π cos x sin d fξe a iξ dξ.6.4. f a x π a cos x sin -. - -.5 - -.5.5.5 d 9 a d π 3.6.3 43 c a ib c fξe +iξ dξ fξcos ξ i sin ξ dξ c 9 a fx e[ce ix ] d π. ce ix + c e ix d a4 π a6 a64 [ d fξe ix ξ dξ π.8 ] + d fξe ix ξ dξ [ d π [ d π + fξe ix ξ dξ dη fξe ix ξ dξ dη fξe ix ξ dξ ] fξe iηx ξ dξ ] fξe iηx ξ dξ x fx + C fxe ix 44 π fx π fx Ce ix d 45 x < π cos x sin π d x C 4 π < x 53 x < i fx x > x sin d π x x < ii fx x > Six x sin d iii fx e x x