Differential Eguations With Boundary-Value Problems SEVENTH EDITION Dennis G.Zill Michael R.Cullen
REVIEW OF DIFFERENTIATION Rules 1.Constant: dc=0 2.Constant Multiple: f()=cf(x) d 3.Sum: f(田±gxl=f'e±gx 4.Product: d dx fxg)=f(g)+gefe 5.Quotient: dfx-gx)f"x)-fx)g'(） dx g(x) 6.Chain: [g(x2 足fgx》=fgxg'(x) 7.Power: d x”=nxn-1 8.Power: dx ng(g( dx Functions Trigonometric: 9. d d sinx=cosx 10. 11. dx dx cosx=-sinx d tanx=seex d 12.4cotx=-csc 13. d 14. dx -secx=sec xtanx cscx=-cscx cot Inverse trigonometric: 15d 1 16. d cos-1x=-- 1 17. dx 1-x2 dx V1-x2 dx 1+x2 18 ot-x=-1 -sec-1x=- 1 19. 20. d csc-1 x=-- 1 dx 1+x2 dx Ve2-1 dx Vx2-1 Hyperbolic: 21 sinhx=coshx 22. d -cosh x=sinhx 23. dx dx d tanhx=sech2x dx 24 -coth x=-csch2x 25. d sechx=-sech xtanhx 26. d -cschx =-csch x cothx dx dx dx Inverse hyperbolic: 27 -sinh-1x=- 1 28. d cosh-1x=- 1 29.d tanhx=1 dx x2+1 dx x2-1 1-x2 30. -coth=7 31. d 1-x2 sech-Ix=- d 32. csch-1x=- xv1-x2 dx Ve2+1 Exponential: mhvve 34.4b=banb) dx Logarithmic: 5县g=生 36. 1 dx d logx= d x(nb)
Rules 1. Constant: d dx c = 0 2. Constant Multiple: d dx cf (x) = c f (x) . Sum: d dx [f (x) ± g(x)] = f (x) ± g(x) 4. Product: d dx f (x)g(x) = f (x)g(x) + g(x) f (x) 5. Quotient: d dx f (x) g(x) = g(x)f (x) f (x)g (x) [ g(x)]2 6. Chain: d dx f (g(x)) = f (g(x))g(x) 7. Power: d dx xn = nxn1 8. Power: d dx [ g(x)]n = n[ g(x)]n 1 g(x) Functions Trigonometric: 9. d dx sin x = cos x 10. d dx cos x = sin x 11. d dx tan x = sec2 x 12. d dx cot x = csc2 x 13. d dx sec x = sec x tan x 14. d dx csc x = csc x cot x Inverse trigonometric: 15. d dx sin1 x = 1 1 x2 16. d dx cos1 x = 1 1 x2 17. d dx tan1 x = 1 1 + x2 18. d dx cot1 x = 1 1 + x2 19. d dx sec1 x = 1 x x2 1 20. d dx csc1 x = 1 x x2 1 Hyperbolic: 21. d dx sinh x = cosh x 22. d dx cosh x = sinh x 23. d dx tanh x = sech2 x 24. d dx coth x = csch2 x 25. d dx sech x = sech x tanh x 26. d dx csch x = csch x coth x Inverse hyperbolic: 27. d dx sinh1 x = 1 x2 +1 28. d dx cosh1 x = 1 x2 1 29. d dx tanh1 x = 1 1 x2 30. d dx coth1 x = 1 1 x2 31. d dx sech1 x = 1 x 1 x2 32. d dx csch1 x = 1 x x2 +1 Exponential: 33. d dx ex = ex 34. d dx bx = bx (lnb) Logarithmic: 35. d dx ln x = 1 x 36. d dx logb x = 1 x(lnb) 3 REVIEW OF DIFFERENTIATION
BRIEF TABLE OF INTEGRALS +C,n≠-1 n+1 2j片=p+c 3∫eda=c+c ∫as 4 a"+C Ina 5.sin udu =-cos u+C 6.cos udu=sin u+C 7. sec2 u du tan u+C 8 csci udu=-cot u+c 9.sec utan udu=sec u+C 10. csc u cot u du =-csc u+C 11.tan udu =-In cos u+C 12. cot udu =In sin u+C 13.sec u du Insec u+tan u+C 14.cse udu Inlesc u-cot u+C 15.u sin udu =sin u-ucosu+C 16.u cos u du cos u+usin u+C 17.sin'udu=tu-sin 2u+C 1 .cos'udu=u+sin u+c 19.tan'udu=tan u-u+C 20. cot2 udu =-cot u-u C 21.sin'udu=(2+sin'u)cosu+C 22. cos'udu=(2+cos'u)sin u+C 23.tan'udu =tan'u+In cos u+C 24 cot'udu =-+cot2u-Insin u+C 25. sec u du =isecu tanu+Insecu+tanu+C 26. cse u du =-+cscu cotu+Incscu-cotu+C 27. sin aucos bu du=sin(a-b)usin(abuC 28 cos au cos bu du-sin(absin(bc 2(a-b) 2(a+b) 2(a-b) 2(a+b) 29. [e"sin budu= e (asin bu-bcos bu)+C 30 e cos bu du = (acos bu+bsin bu)+C 31. sinh u du cosh u+C 32. cosh u du sinh u+C 33. sech2 u du tanh u+C 34. csch2 u du =-coth u+C 35. tanh u du In(cosh u)+C 36. coth u du Insinh u+C 37. In udu =ulnu-u+C 38. uln udu =iu2 Inu-iu2+C 39. a2- -du=sin+C 40. a ∫-=-r+ sin+C 42. ∫F+-F+++F++c 43. du=-tan+C 44. a2-= 1a+4+C Ja2+u a 2aa-u Note:Some techniques of integration,such as integration by parts and partial fractions,are reviewed in the Student Resource and Solutions Manal that accompanies this text
BRIEF TABLE OF INTEGRALS 1. 1 , 1 1 n n u u du C n n 2. 1 du u C ln u 3. u u e du e C 4. 1 ln u u a du a C a 5. sin cos u du u C 6. cos sin u du u C 7. 2 sec tan u du u C 8. 2 csc cot u du u C 9. sec tan sec u u du u C 10. csc cot csc u u du u C 11. tan ln cos u du u C 12. cot ln sin u du u C 13. sec ln sec tan u du u u C 14. csc ln csc cot u du u u C 15. u u du u u u C sin sin cos 16. u u du u u u C cos cos sin 17. 2 1 1 2 4 sin sin 2 u du u u C 18. 2 1 1 2 4 cos sin 2 u du u u C 19. 2 tan tan u du u u C 20. 2 cot cot u du u u C 21. 3 2 1 3 sin 2 sin cos u du u u C 22. 3 2 1 3 cos 2 cos sin u du u u C 23. 3 2 1 2 tan tan ln cos u du u u C 24. 3 2 1 2 cot cot ln sin u du u u C 25. 3 1 1 2 2 sec sec tan ln sec tan u du u u u u C 26. 3 1 1 2 2 csc csc cot ln csc cot u du u u u u C 27. sin( ) sin( ) sin cos 2( ) 2( ) a bu a bu au bu du C ab ab 28. sin( ) sin( ) cos cos 2( ) 2( ) a bu a bu au bu du C ab ab 29. 2 2 sin sin cos au au e e bu du a bu b bu C a b 30. 2 2 cos cos sin au au e e bu du a bu b bu C a b 31. sinh cosh u du u C 32. cosh sinh u du u C 33. 2 sech tanh u du u C 34. 2 csch coth u du u C 35. tanh ln(cosh ) u du u C 36. coth ln sinh u du u C 37. ln ln u du u u u C 38. 1 1 2 2 2 4 u u du u u u C ln ln 39. 1 2 2 1 sin u du C a u a 40. 2 2 2 2 1 du u a u C ln a u 41. 2 22 22 1 sin 2 2 u au a u du a u C a 42. 2 22 22 22 ln 2 2 u a a u du a u u a u C 43. 1 2 2 1 1 tan u du C a u a a 44. 2 2 1 1 ln 2 a u du C a u a au Note: Some techniques of integration, such as integration by parts and partial fractions, are reviewed in the Student Resource and Solutions Manual that accompanies this text.
TABLE OF LAPLACE TRANSFORMS f(r) {f0=F) f() f())=F(s) 1.1 20.eat sinh k (s a)-K 21 21.ea cosh kt s-a 8-a2-2 3. n! napositive integer 22.tsin kt 2ks 2+2乎 4.1~2 限 23.tcoskt 2-k2 2+k2乎 5.72 温 24.sin kt kt cos kt 2k2 2+ 6. 25.sin kt-kt cos kt 2 G2+2乎 7.sin kt 2ks 2+2 26.t sinh kt 2-2呼 8.cos kt 27.t cosh kt s2+k2 2+2 (G2-k乎 22 28-山 1 9.sin2kt 2+4码 a-b (s-0s-b) 10.cos2kt 2+22 sG2+4k码 29.ae-beh a-b (s-a)(s -b) 1 2 11.eat s-a 30.1-cos kt s(2+码 3 12.sinh kt 2-R 3L.k红-sinkt 22+巧 13.cosh kt 32.asin br-bsin at ab(a2-b2) 2+232+b 14.sinh2kt 22 s62-4码 33.cos bi-cos at a2-b2 2+a2s2+b的 15.cosh2kr 32-2k2 34.sin kt sinh kr 23 s(2-4k码 ++40 1 16.tea 35.sin kt cosh kt k(2+2k2) 6- +4 17."eat n! 36.cos kt sinh kt k2-2 (s -ayi, n a positive integer s4+4 k 18.ett sin kt 6-a2+F 37.cos kt cosh kr +4 19.eat cos kt s-a G-a2+k及 38.Jo) 1 VF+元
TABLE OF LAPLACE TRANSFORMS f(t) 1. 1 2. t 3. t n n a positive integer 4. t 1/2 5. t 1/2 6. t a 7. sin kt 8. cos kt 9. sin2 kt 10. cos2 kt 11. eat 12. sinh kt 13. cosh kt 14. sinh2kt 15. cosh2kt 16. teat 17. t n eat n a positive integer 18. eat sin kt 19. eat cos kt s a (s a) 2 k2 k (s a) 2 k2 n! (s a) n1 , 1 (s a) 2 s2 2k2 s(s2 4k2 ) 2k2 s(s2 4k2 ) s s2 k2 k s2 k2 1 s a s2 2k2 s(s2 4k2 ) 2k2 s(s2 4k2 ) s s2 k2 k s2 k2 ( 1) s1 , a 1 1 2s3/2 B s n! sn1 , 1 s2 1 s { f (t)} F(s) f(t) 20. eat sinh kt 21. eat cosh kt 22. t sin kt 23. t cos kt 24. sin kt kt cos kt 25. sin kt kt cos kt 26. t sinh kt 27. t cosh kt 28. 29. 30. 1 cos kt 31. kt sin kt 32. 33. 34. sin kt sinh kt 35. sin kt cosh kt 36. cos kt sinh kt 37. cos kt cosh kt 38. J0(kt) 1 1s2 k2 s3 s4 4k4 k(s2 2k2) s4 4k4 k(s2 2k2) s4 4k4 2k2 s s4 4k4 s (s2 a2 )(s2 b2 ) cos bt cos at a2 b2 1 (s2 a2 )(s2 b2 ) a sin bt b sin at ab (a2 b2 ) k3 s2(s2 k2 ) k2 s(s2 k2 ) s (s a)(s b) aeat bebt a b 1 (s a)(s b) eat ebt a b s2 k2 (s2 k2 ) 2 2ks (s2 k2 ) 2 2k3 (s2 k2 ) 2 2ks2 (s2 k2 ) 2 s2 k2 (s2 k2 ) 2 2ks (s2 k2 ) 2 s a (s a) 2 k2 k (s a) 2 k2 { f (t)} F(s)
SEVENTH EDITION DIFFERENTIAL EQUATIONS with Boundary-Value Problems
SEVENTH EDITION DIFFERENTIAL EQUATIONS with Boundary-Value Problems