# Differential Equations with Boundry Value Problems Essay

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REVIEW OF DIFFERENTIATION Rules 1. Constant: d c=0 dx 2. Constant Multiple: 4. Product: d cf (x) = c f (x) dx 3 . Sum: d [ f (x) ± g(x)] = f (x) ± g (x) dx d f (x) g(x)f (x) f (x) g (x) = 5. Quotient: dx g(x) [ g(x)]2 7. Power: d f (x) g(x) = f (x) g (x) + g(x) f (x) dx d 6. Chain: f ( g(x)) = f ( g(x)) g (x) dx 8. Power: d n x = nx n dx 1 d [ g(x)]n = n[ g(x)]n dx 1 g (x) Functions Trigonometric: d 9. sin x = cos x dx d cot x = csc 2 x 12. dx Inverse trigonometric: d 1 15. sin 1 x = dx 1 x2 18. d cot dx 1 10. d cos x = sin x dx d 13. sec x = sec x tan x dx 16. 19. d cos dx d sec dx 1 11. d tan x = sec 2 x dx d 14. csc x = csc x cot x dx d tan dx d csc dx 1 x= x= x 1 1 x 1 x 2 2 17. 20. x= x= 1 1 + x2 1 x x2 1 x= 1 1+ x 2 1 1 1 Hyperbolic: d 21. sinh x = cosh x dx d coth x = csch 2 x 24. dx Inverse hyperbolic: 27. 30. d sinh dx d coth dx 1 22. d cosh x = sinh x dx d 25. sech x = sech x tanh x dx d cosh dx d sech dx 1 23. d tanh x = sech 2 x dx d 26. csch x = csch x coth x dx 29. d tanh dx d csch dx 1 x= x= 1 x +1 1 1 x 2 2 28. 31. x= x x= 1 2 x= x= 1 1 x x2 1 x2 + 1 1 1 1 1 x2 32. 1 x 1 Exponential: d x e = ex 33. dx Logarithmic: 35. 1 d ln x = x dx 34. d x b = bx (ln b) dx d 1 log b x = dx x(ln b) 36. BRIEF TABLE OF INTEGRALS 1. 3. 5. 7. 9. 11. 13. 15. 17. 19. 21. 23. 25. 27. 29. 31. 33. 35. 37. 39. 41. 43. ³ ³ e du e  C ³ sin u du  cos u  C ³ sec u du tan u  C ³ sec u tan u du sec u  C ³ tan u du  ln cos u  C ³ sec u du ln sec u  tan u  C ³ u sin u du sin u  u cos u  C ³ sin u du u  sin 2u  C ³ tan u du tan u  u  C ³ sin u du  2  sin u cos u  C ³ tan u du tan u  ln cos u  C ³ sec u du sec u tan u  ln sec u  tan u  C sin( a  b)u sin( a  b)u ³ sin au cos bu du 2(a  b) 