Engineering math formula for GATE/BARC/ISRO

Engineering Mathematics is one of the scoring sections in GATE/BARC/ISRO Exam. Looking at your requirement, we are sharing with you Important Engineering Mathematics Formulas & Shortcuts for Competitive Exam as well as Engineering exam.

1. If 𝑟≠𝑟′, no solution, if 𝑟=𝑟′=𝑛, unique solution if 𝑟=𝑟′<𝑛, many solutions. (non-homogeneous)
2. If 𝑟=𝑛, trivial solution, if 𝑟<𝑛 ,then (𝑛−𝑟) linearly independent solutions. (Many solutions) and if 𝑚<𝑛, then many solutions.
3. 𝑓(𝑥+ℎ) =𝑓(𝑥) +ℎ𝑓′(𝑥) +ℎ22!𝑓′′(𝑥) +ℎ33!𝑓′′′(𝑥) + ………∞
4. If 𝑟𝑡−𝑠2>0 and 𝑟<0 𝑓(𝑥,𝑦) have maximum, if 𝑟𝑡−𝑠2>0 and 𝑟>0 𝑓(𝑥,𝑦) have minimum at(𝑎,𝑏) and if 𝑟𝑡−𝑠2<0, then saddle point. If 𝑟𝑡−𝑠2=0,𝑓𝑢𝑟𝑡ℎ𝑒𝑟 investigation is required to decide.
5. ∫𝑐(∅𝑑𝑥+𝜓𝑑𝑦)=∫∫𝐸(𝜕𝜓𝜕𝑥−𝜕∅𝜕𝑦)𝑑𝑥𝑑𝑦 (Green’s)
6. ∫𝑐𝑭.𝑑ℝ=∫𝑆𝑐𝑢𝑟𝑙𝑭.𝑁𝑑𝑠 (Stokes)
7. ∫𝑆𝑭.𝑁𝑑𝑠=∫𝐸𝑑𝑖𝑣𝑭𝑑𝑣 (Gauss)
8. 𝑦𝑒∫𝑃𝑑𝑥=∫𝑄𝑒∫𝑃𝑑𝑥𝑑𝑥+𝑐
9. If 𝑀 𝑑𝑥+𝑁 𝑑𝑦=0 be a homogeneous equation in 𝑥 and 𝑦, then 1𝑀 𝑥+𝑁 𝑦 𝑖𝑠 an integrating factor
10. If the equation of the type 𝑓1(𝑥𝑦)𝑦 𝑑𝑥+𝑓2(𝑥𝑦)𝑥 𝑑𝑦=0. If the equation 𝑀 𝑑𝑥+𝑁 𝑑𝑦=0 be of this type then 1𝑀 𝑥−𝑁 𝑦 is an integrating factor
11. If 𝜕𝑀𝜕𝑦−𝜕𝑁𝜕𝑥𝑁 be a function of x only = 𝑓(𝑥) say then 𝑒∫𝑓(𝑥)𝑑𝑥 is an integrating factor

12. If 𝜕𝑁𝜕𝑥−𝜕𝑀𝜕𝑦𝑀 be a function of y only = 𝑓(𝑦) say then 𝑒∫𝑓(𝑦)𝑑𝑦 is an integrating factor.
13. ∫𝑀𝑑𝑥𝑦=𝑐𝑜𝑛𝑡 +∫terms of N not containing x dy=c
14. 𝑃.𝐼.=1𝑓(𝐷)𝑒𝑎𝑥=1𝑓(𝑎)𝑒𝑎𝑥 , 𝑓(𝑎)≠0, if 𝑓(𝑎)=0,𝑡ℎ𝑒𝑛 𝑃.𝐼.=𝑥1𝑓′(𝑎)𝑒𝑎𝑥,𝑓′(𝑎)≠0
15. 𝑃.𝐼=1𝑓(𝐷2)sin(𝑎𝑥+𝑏)= 1𝑓(−𝑎2), 𝑓(−𝑎2)≠0, if 𝑓(−𝑎2)=0,

then 𝑃.𝐼. =𝑥1𝑓′(−𝑎2)sin(𝑎𝑥+𝑏), 𝑓′(−𝑎2)≠0

16. 𝑃.𝐼.=1𝑓(𝐷)𝑒𝑎𝑥𝑉= 𝑒𝑎𝑥1𝑓(𝐷+𝑎)𝑉
17. 𝑃.𝐼=1𝑓(𝐷)𝑥𝑚=[𝑓(𝐷)]−1𝑥𝑚,
18. (1+𝑥)−1=1−𝑥+𝑥2−⋯
19. (1−𝑥)−1=1+𝑥+𝑥2+⋯
20. 𝑥𝑛𝑑𝑛𝑦𝑑𝑥𝑛+𝑘1𝑥𝑛−1𝑑𝑛−1𝑦𝑑𝑥𝑛−1+⋯𝑘𝑛−1𝑥𝑑𝑦𝑑𝑥+𝑘𝑛𝑦=𝑋, 𝑥=𝑒𝑡 , 𝑥𝑑𝑦𝑑𝑥=𝐷𝑦, 𝑥2𝑑2𝑦𝑑𝑥2=𝐷(𝐷−1)𝑦, 𝑥3𝑑3𝑦𝑑𝑥3=D(D−1)(D−2)
21. 𝜕𝜕𝑥(∫ℎ(𝑡,𝑥)𝑑𝑡𝑔(𝑥)𝑓(𝑥))=∫𝜕𝜕𝑥ℎ(𝑡,𝑥)𝑑𝑡𝑔(𝑥)𝑓(𝑥)+𝑑𝑔𝑑𝑥 ℎ[𝑔(𝑥),𝑥]−𝑑𝑓𝑑𝑥 ℎ[𝑓(𝑥),𝑥]
22. 𝐿{𝑓(𝑡)}=∫𝑒−𝑠𝑡∞0 𝑓(𝑡)𝑑𝑡
23. 𝐿(1)=1𝑠
24. 𝐿(𝑡𝑛)=𝑛!𝑠𝑛+1
25. 𝐿(𝑒𝑎𝑡)=1𝑠−𝑎
26. 𝐿(sin𝑎𝑡)=𝑎𝑠2+𝑎2
27. 𝐿(cos𝑎𝑡)=𝑠𝑠2+𝑎2
28. 𝐿(sinh𝑎𝑡)=𝑎𝑠2−𝑎2
29. 𝐿(cosh𝑎𝑡)=𝑠𝑠2−𝑎2
30. 𝐿{𝑒𝑎𝑡𝑓(𝑡)}= 𝑓̅(𝑠−𝑎)
31. 𝑓(𝑡+𝑇)=𝑓(𝑡) then 𝐿{𝑓(𝑡)}= ∫𝑒−𝑠𝑡 𝑓(𝑡)𝑑𝑡𝑇01−𝑒−𝑠𝑇
32. 𝐿{𝑓′(𝑡)}=𝑠𝑓̅(𝑠)−𝑓(0)
33. 𝐿{𝑓𝑛(𝑡)}= 𝑠𝑛𝑓̅(𝑠)−𝑠𝑛−1𝑓(0)−𝑠𝑛−2𝑓′(0)−⋯……..𝑓𝑛−1(0)
34. 𝐿{∫𝑓(𝑥)𝑑𝑥𝑡0}= 1𝑆𝑓̅(𝑠)
35. 𝐿{𝑡𝑛𝑓(𝑡)}=(−1)𝑛𝑑𝑛𝑑𝑠𝑛.[𝑓̅(s)]
36. 𝐿{1𝑡𝑓(𝑡)}= ∫𝑓̅(s) ∞𝑆𝑑𝑠
37. 𝑓(𝑥)=𝑎02+ Σ𝑎𝑛∞𝑛=1cos𝑛𝑥+ Σ𝑏𝑛∞𝑛=1sin𝑛𝑥
38. 𝑎0=1𝜋∫𝑓(𝑥)𝑑𝑥𝛼+2𝜋𝛼, 𝑎𝑛=1𝜋∫𝑓(𝑥) cos𝑛𝑥𝑑𝑥𝛼+2𝜋𝛼, 𝑏𝑛=1𝜋∫𝑓(𝑥) sin𝑛𝑥𝑑𝑥𝛼+2𝜋𝛼
39. 𝑓(𝑥)=𝑎02+ Σ𝑎𝑛∞𝑛=1cos𝑛𝜋𝑥𝑐+ Σ𝑏𝑛∞𝑛=1sin𝑛𝜋𝑥𝑐
40. 40. 𝑎0=1𝑐∫𝑓(𝑥)𝑑𝑥𝛼+2𝑐𝛼, 𝑎𝑛=1𝑐∫𝑓(𝑥) cos𝑛𝜋𝑥𝑐𝑑𝑥𝛼+2𝑐𝛼, 𝑏𝑛=1𝑐∫𝑓(𝑥) sin𝑛𝜋𝑥𝑐𝑑𝑥𝛼+2𝑐𝛼
41. 𝑓(𝑥)= Σ𝑏𝑛∞𝑛=1sin𝑛𝜋𝑥𝑐 , where 𝑏𝑛=2𝑐∫𝑓(𝑥) sin𝑛𝜋𝑥𝑐𝑑𝑥𝑐0
42. 𝑓(𝑥)= 𝑎02+ Σ𝑎𝑛∞𝑛=1cos𝑛𝜋𝑥𝑐 where, 𝑎0=2𝑐∫𝑓(𝑥)𝑑𝑥𝑐0, 𝑎𝑛=2𝑐∫𝑓(𝑥) cos𝑛𝜋𝑥𝑐𝑑𝑥𝑐0
43. 𝜇=Σ 𝑥𝑗𝑓(𝑥𝑗)𝑗 𝑎𝑛𝑑 𝜇=∫𝑥 𝑓(𝑥)𝑑𝑥 ∞−∞
44. 𝜎2=Σ (𝑥𝑗−𝜇)2𝑓(𝑥𝑗) 𝑗𝑎𝑛𝑑 𝜎2=∫(𝑥−𝜇)2 𝑓(𝑥)𝑑𝑥∞−∞
45. 𝑀𝑒𝑎𝑛:𝑛𝑝=𝜇 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒: 𝜎2=𝜇 (Poisson’s distribution)
46. 𝑓(𝑥)=1𝜎√2𝜋𝑒− 12(𝑥−𝜇𝜎)2 (Normal distribution)
47. 𝑦=𝑎+𝑏𝑥, Σ𝑦=𝑛𝑎+𝑏Σ𝑥, Σ𝑥𝑦=𝑎Σ𝑥+𝑏Σ𝑥2
48. 𝑦=𝑎+𝑏𝑥+𝑐𝑥2, Σ𝑦=𝑛𝑎+𝑏.Σ𝑥+𝑐.Σ𝑥2, Σ𝑥𝑦=𝑎Σ𝑥+𝑏.Σ𝑥2+𝑐 Σ𝑥3, Σ𝑥2𝑦=𝑎Σ𝑥2+𝑏.Σ𝑥3+𝑐 Σ𝑥4
49. 𝑦=𝑓(𝑥)=(𝑥−𝑥1)(𝑥−𝑥2)…(𝑥−𝑥𝑛)(𝑥0−𝑥1)(𝑥0−𝑥2)…(𝑥0−𝑥𝑛)𝑦0+(𝑥−𝑥0)(𝑥−𝑥2)…(𝑥−𝑥𝑛)(𝑥1−𝑥0)(𝑥1−𝑥2)…(𝑥1−𝑥𝑛)𝑦1+⋯+(𝑥−𝑥1)(𝑥−𝑥2)…(𝑥−𝑥𝑛−1)(𝑥𝑛−𝑥0)(𝑥𝑛−𝑥1)…(𝑥𝑛−𝑥𝑛−1)𝑦𝑛
50. (𝑑𝑦𝑑𝑥)𝑥0=1ℎ[Δ𝑦0−12Δ2𝑦0+13Δ3𝑦0−14Δ4𝑦0+⋯]
51. (𝑑𝑦𝑑𝑥)𝑥𝑛=1ℎ[∇𝑦𝑛+12∇2𝑦𝑛+13∇3𝑦𝑛+14∇4𝑦𝑛+⋯]
52. 𝑥𝑛+1=𝑥𝑛−𝑓(𝑥𝑛)𝑓′(𝑥𝑛) (Newton-Raphson)
53. ∫𝑓(𝑥)𝑑𝑥𝑥0+𝑛ℎ𝑥0=ℎ2[𝑦0+𝑦𝑛+2(𝑦1+𝑦2+⋯..+𝑦𝑛−1)] (Trapezoidal)
54. ∫𝑓(𝑥)𝑑𝑥𝑥0+𝑛ℎ𝑥0=ℎ3[(𝑦0+𝑦𝑛)+4(𝑦1+𝑦3+⋯𝑦𝑛−1)+2(𝑦2+𝑦4+⋯𝑦𝑛−2)] (Simpson’s)
55. 𝐸𝑟𝑟𝑜𝑟=−𝑏−𝑎12ℎ2 𝑓′′(𝜉)=𝑂(ℎ2) (Trapezoidal)
56. 𝐸𝑟𝑟𝑜𝑟=−𝑏−𝑎180ℎ4𝑓𝑖𝑣(𝜉)=𝑂(ℎ4) (Simpson’s)
57. 𝑦𝑘+1=𝑦𝑘+ℎ.𝑓(𝑡𝑘,𝑦𝑘) where 𝑑𝑦𝑑𝑥=𝑓(𝑡,𝑦) (Euler’s)