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- 2. Differential Equations ❖ TeachingExperience: 10+ Years ❖ Qualification: M.Tech from NIT Bhopal ❖ Achievements: ▪ Cracked GATE Many Times with Good GATE Score & rank ▪ Written Many Research and Review Papers ❖ Area of Expertise: ▪ Strength of Materials, Theory of Machines, Machine Design, Industrial Engineering & Production. DHEERAJ SARDANA ABOUT ME
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- 5. Differential Equations Find Linkin the Description.
- 6. Differential Equations Find Linkin the Description.
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- 8. Differential Equations Join MyTelegram Group https://t.me/dheeraj_sardana ✓ For daily dose to improve concepts ✓ For daily important questions ✓ For latest information about exams ✓ For notification of the class ✓ For important PDFs & notes ✓ For clearing the doubts by experts DHEERAJ SARDANA
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- 10. Differential Equations Find Linkin the Description.
- 11. Differential Equations Order °ree of Differential equation: The order of highest ordered derivative occurring in D. E. is known as Order of D.E Degree of a D.E. is highest ordered derivative occurring in it when derivatives are free from fractional powers.
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- 13. Differential Equations Q. Thepartial differential equation 𝑑2∅ 𝑑𝑥2 + 𝑑2∅ 𝑑𝑦2 + 𝑑∅ 𝑑𝑥 + 𝑑∅ 𝑑𝑥 = 0 has (a)Degree 1 order 2 (b) degree 1 order 1 (c) Degree 2 order 1 (d) degree 2 order 2
- 14. Differential Equations Solution ofOrdinary Differential Equations First Method Variable separable (I) 𝑑𝑦 𝑑𝑥 = 𝑋 𝑥 𝑌 𝑦 → 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑥 𝑜𝑛𝑙𝑦 → 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑦 𝑜𝑛𝑙𝑦 or 𝑑𝑦 𝑑𝑥 = 𝑋 𝑥 ∙ 𝑌 𝑦
- 15. Differential Equations Q. Thesolution of the differential equation 𝑑𝑦 𝑑𝑥 + 𝑦2=0 is (a) y= 1 𝑥+𝑐 (b) 𝑦 = −𝑥3 3 + 𝑐 (c) c𝑒𝑥 (d) Unsolvable as equation is non-linear
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- 17. Differential Equations Solution ofOrdinary Differential Equations (II) 𝑑𝑦 𝑑𝑥 = 𝑓 𝑎𝑥 + 𝑏𝑦 + 𝑐
- 18. Differential Equations Q. 𝑑𝑦 𝑑𝑥 = 𝑥+ 𝑦 + 1 2
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- 20. Differential Equations Q. 𝑑𝑖 𝑑𝑡 − 2𝑖= 0 is applicable over −10 < 𝑡 < 10 if 𝑖 4 = 10 then 𝑖 −5 is
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- 22. Differential Equations Q. forD.E. 𝑑𝑦 𝑑𝑡 + 5𝑡 = 0 𝑦 0 = 1 find solution. A. 𝑒5𝑡 B. 𝑒−5𝑡 C. 5𝑒−5𝑡 D. 𝑒 −5𝑡
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- 24. Differential Equations Q. Ifat every point of a certain curve the slope of the tangent −2𝑥 𝑦 the curve is A. straight line B. parabola C. circle D. ellipse
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- 26. Differential Equations Q. Solutionof first order D.E. ሶ 𝑥 𝑡 = −3𝑥 𝑡 𝑥 0 = 𝑥0 is A. 𝑥 𝑡 = 𝑥0 𝑒−3𝑡 B. 𝑥 𝑡 = 𝑥0 𝑒−3 C. 𝑥 𝑡 = 𝑥0 𝑒−𝑡/3 D. 𝑥 𝑡 = 𝑥0 𝑒−𝑡
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- 28. Differential Equations Q. Thesolution of 𝑑𝑦 𝑑𝑥 = 𝑦2 with initial value 𝑦 0 = 1 bounded in this interval is A. −∞ ≤ 𝑥 ≤ ∞ B. −∞ ≤ 𝑥 ≤ 1 C. 𝑥 < 1, 𝑥 > 1 D. −2 ≤ 𝑥 ≤ 2
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- 30. Differential Equations Q. Thesolution of differential equation 𝑑𝑦 𝑑𝑥 − 𝑦2 = 1 satisfying condition 𝑦 0 = 1 is A. 𝑦 = 𝑒𝑥2 B. 𝑦 = cot 𝑥 + 𝜋 4 C. 𝑦 = 𝑥 D. 𝑦 = tan 𝑥 + 𝜋 4
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- 32. Differential Equations Q. Matchthe following P: 𝑑𝑦 𝑑𝑥 = 𝑦 𝑥 Q : 𝑑𝑦 𝑑𝑥 = − 𝑦 𝑥 R : 𝑑𝑦 𝑑𝑥 = 𝑥 𝑦 S : 𝑑𝑦 𝑑𝑥 = − 𝑥 𝑦 (1) Circle (2) St. line (3) Hyperbola
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- 34. Differential Equations Q. Thegeneral solution of 𝑑𝑦 𝑑𝑥 = cos 𝑥 + 𝑦 with 𝑐 as constant A. 𝑦 + sin(𝑥 + 𝑦) B. cos 𝑥+𝑦 2 = 𝑥 + 𝑐 C. tan 𝑥+𝑦 2 = 𝑦 + 𝑐 D. tan 𝑥+𝑦 2 = 𝑥 + 𝑐
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- 36. Differential Equations Q. Whichone of the following is the general solution of 1st order D.E. 𝑑𝑦 𝑑𝑥 = 𝑥 + 𝑦 − 1 2; 𝑥 & 𝑦 are real A. 𝑦 = 1 + 𝑥 + tan−1 (𝑥 + 𝑐) 𝑐 is constant B. 𝑦 = 1 + 𝑥 + tan 𝑥 + 𝑐 𝑐 is constant C. 𝑦 = 1 − 𝑥 + tan−1 𝑥 + 𝑐 𝑐 is constant D. 𝑦 = 1 − 𝑥 + tan 𝑥 + 𝑐 𝑐 is constant
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- 38. Differential Equations Linear DifferentialEQN (I) 𝑑𝑦 𝑑𝑥 + 𝑃𝑦 = 𝑄 𝑃 & 𝑄 = FNS of 𝑥 only
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- 40. Differential Equations Linear DifferentialEQN (II) 𝑑𝑦 𝑑𝑥 + 𝑃𝑥 = 𝑄 𝑃 & 𝑄 = FNS of 𝑦 only
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- 42. Differential Equations Q. If𝑥2 ∙ 𝑑𝑦 𝑑𝑥 + 2𝑥𝑦 = 2 ln 𝑥 𝑥 𝑦 1 = 0 find 𝑦(𝑒) A. 𝑒 B. 1 C. 1/𝑒 D. 1/𝑒2
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- 44. Differential Equations Q. 𝑑𝑦 𝑑𝑥 + 2𝑥𝑦= 𝑒−𝑥2 with 𝑦 0 = 1 A. 1 + 𝑥 𝑒𝑥2 B. 1 + 𝑥 𝑒−𝑥2 C. 1 − 𝑥 𝑒𝑥2 D. 1 − 𝑥 𝑒−𝑥2
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- 46. Differential Equations Q. Thesolution of the differential equation 𝑑𝑦 𝑑𝑥 + 2𝑥𝑦 = 𝑒(−𝑥2) with y(0) = 1 is (a) (1 + x) 𝑒(+𝑥2) (b) (1 + x) 𝑒(−𝑥2) (c) (1 - x) 𝑒(+𝑥2) (d (1 - x) 𝑒(−𝑥2)
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- 48. Differential Equations Q. 𝑥. 𝑑𝑦 𝑑𝑥 +𝑦 = 𝑥4 𝑦 1 = 6 5 A. 𝑦 = 𝑥4 5 + 1 𝑥 B. 𝑦 = 4𝑥4 5 + 4 5𝑥 C. 𝑦 = 𝑥4 5 + 1 D. 𝑦 = 𝑥5 5 + 1
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- 50. Differential Equations Homogeneous DifferentialEquations :- 𝑑𝑦 𝑑𝑥 = 𝑓(𝑥, 𝑦) 𝑔 𝑥, 𝑦 → both are homogeneous → function 𝑓 𝑥, 𝑦 = 𝑥3 + 𝑦3 + 𝑥2𝑦 → power of each term is same Condition 𝑓 𝑘𝑥, 𝑘𝑦 = 𝑘𝑛𝑓 𝑥, 𝑦 ⇒ Homogeneous 𝑓 𝑘𝑥, 𝑘𝑦 ≠ 𝑘𝑛𝑓 𝑥, 𝑦 ⇒ Non-Homogeneous
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- 52. Differential Equations Q. Acurve passing through a point 1, 𝜋 6 Let the slope of curve at each point (𝑥, 𝑦) be 𝑦 𝑥 + sec 𝑦 𝑥 . Then eqn. of curve 𝑥 > 0 A. sin 𝑦 𝑥 = ln 𝑥 + 1 2 B. cos 𝑦 𝑥 = log 𝑥 + 1 2 C. sin 2𝑦 𝑥 = log 𝑥 + 2 D. cos 2𝑦 𝑥 = log 𝑥 + 1 2
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- 54. Differential Equations Second orderlinear diff. Eqn : 𝑑𝑛𝑦 𝑑𝑥𝑛 + 𝑃𝑑𝑛−1𝑦 𝑑𝑥𝑛−1 + 𝑄𝑑𝑛−2𝑦 𝑑𝑥𝑛−2 + ⋯ + 𝑡𝑦 = 𝑅 𝑥 𝑃 & 𝑄 are constant for second order 𝑛 = 2 𝑑2 𝑦 𝑑𝑥2 + 𝑃𝑑𝑦 𝑑𝑥 + 𝑄𝑦 = 𝑅 𝑥
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- 56. Differential Equations Homogeneous linearDE 𝑑2𝑦 𝑑𝑥2 + 𝑃𝑑𝑦 𝑑𝑥 + 𝑄𝑦 = 0 𝑃 & 𝑄 are constant
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- 58. Differential Equations Q. 𝑑2𝑦 𝑑𝑥2 −5 𝑑𝑦 𝑑𝑥 + 6 = 0
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- 60. Differential Equations Case 2: If roots are real & equal C.F. = 𝑐1 + 𝑐2𝑥 𝑒𝑟𝑥 four roots are equal 𝐶. 𝐹. = 𝑐1 + 𝑐2𝑥 + 𝑐3𝑥2 + 𝑐4𝑥3 𝑒𝑟𝑥
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- 62. Differential Equations Case 3: If roots are complex 𝑧 = 𝑎 ± 𝑖𝑏 𝑦 = 𝐶. 𝐹. = 𝑒𝑎𝑥 𝑐1 cos 𝑏𝑥 + 𝑐2 sin 𝑏𝑥
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- 66. Differential Equations Q. Givenan ordinary D.E. 𝑑2𝑦 𝑑𝑥2 + 𝑑𝑦 𝑑𝑥 − 6𝑦 = 0 𝑦 0 = 0 𝑑𝑦 𝑑𝑥 0 = 1 Find value of 𝑦(1) :
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- 68. Differential Equations Q. Themaximum value of solution 𝑦 𝑡 of D.E. 𝑦 𝑡 + 𝑦″(𝑡) = 0 with initial condition 𝑦′ 0 = 1 & 𝑦 0 = 1 for 𝑡 ≥ 0 is A. 1 B. 2 C. 𝜋 D. 2
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- 70. Differential Equations Q. Afunction 𝑛(𝑥) ssatisfies the differential eqn 𝑑2𝑛 𝑥 𝑑𝑥2 − 𝑛 𝑥 𝐿2 = 0 where 𝐿 is a constant The boundary conditions are 𝑛 0 = 𝐾 & 𝑛 ∞ = 0 The soln to eqn is - A. 𝑛 𝑥 = −𝐾 𝑒−𝑥/𝐿 B. 𝑛 𝑥 = 𝐾 𝑒−𝑥/ 𝐿 C. 𝑛 𝑥 = 𝐾 𝑒−𝑥/𝐿 D. 𝑛 𝑥 = 𝐾2 𝑒−𝑥/ 𝐿
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- 72. Differential Equations Particular Integral: 𝑑2 𝑦 𝑑𝑥2 + 𝑃𝑑𝑦 𝑑𝑥 + 𝑄 = 𝑅 𝑥 𝑅(𝑥) is a function of 𝑥 only → 𝑅 𝑥 = 𝑒𝑎𝑥+𝑏 → 𝑅 𝑥 = 𝑥𝑚 → 𝑅 𝑥 = sin(𝑎𝑥 + 𝑏) or cos(𝑎𝑥 + 𝑏)
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- 74. Differential Equations Case 1: 𝑅𝑥 = 𝑒𝑎𝑥+𝑏 𝑑2𝑦 𝑑𝑥2 + 2𝑑𝑦 𝑑𝑥 + 6𝑦 = 𝑒2𝑥+1
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- 78. Differential Equations Case 2: 𝑅𝑥 = sin(𝑎𝑥 + 𝑏) or cos(𝑎𝑥 + 𝑏) 𝑑2𝑦 𝑑𝑥2 + 2𝑑𝑦 𝑑𝑥 + 𝑦 = sin 2𝑥 + 1 or cos(2𝑥 + 1)
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- 80. Differential Equations Case 3: 𝑑2 𝑦 𝑑𝑥2 + 𝑃𝑑𝑦 𝑑𝑥 +𝑄𝑦 = 𝑥𝑚 → algebraic function 𝐷2𝑦 + 𝑃 𝐷𝑦 + 𝑄𝑦 = 𝑥𝑚
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- 82. Differential Equations Q. Thesolution of D.E. 𝑑2𝑦 𝑑𝑥2 + 6𝑑𝑦 𝑑𝑥 + 9𝑦 = 9𝑥 + 6 𝑐1 & 𝑐2 = constants A. 𝑐1𝑥 + 𝑐2 𝑒−3𝑥 B. 𝑐1𝑥 + 𝑐2 𝑒−3𝑥 + 𝑥 C. 𝑐1 𝑒3𝑥 + 𝑐2 𝑒−3𝑥 D. 𝑐1𝑥 + 𝑐2 𝑒3𝑥 + 𝑥
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- 86. Differential Equations Given thatሷ 𝑥 + 3𝑥 = 0, 𝑎𝑛𝑑 𝑥 0 = 1, ሶ 𝑥 0 = 0, what is x(1)? (a) -0.99 (b) -0.16 (c) 0.16 (d) 0.99
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- 88. Differential Equations Q. Considerthe differential equation 𝑑𝑦 𝑑𝑥 = 1 + 𝑦2 𝑥. The general solution with constant c is (a) y = tan 𝑥2 2 + tan 𝑐 (b) 𝑦 = 𝑡𝑎𝑛2 𝑥 2 + 𝑐 (c) 𝑦 = 𝑡𝑎𝑛2 𝑥 2 +c (d) y = tan 𝑥2 2 + 𝑐
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- 90. Differential Equations Consider thedifferential equation 𝑥2 𝑑2𝑦 𝑑𝑥2 + 𝑥 𝑑𝑦 𝑑𝑥 − 4𝑦 = 0 with the boundary conditions of y(0) = 0 and y(1) = 1 The complete solution of the differential equation is (a) 𝑥2 (b) sin 𝜋𝑥 2 (c) 𝑒𝑥 sin 𝜋𝑥 2 (d) 𝑒−𝑥 sin 𝜋𝑥 2
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- 92. Differential Equations Find thesolution of 𝑑2𝑦 𝑑𝑥2 = 𝑦 which passes through the origin and the point (ln2, 3 4 ) (a) 𝑦 = 1 2 𝑒𝑥- 𝑒−𝑥 (b)𝑦 = 1 2 (𝑒𝑥 + 𝑒−𝑥 ) (c) 𝑦 = 1 2 (𝑒𝑥 − 𝑒−𝑥) (d)𝑦 = 1 2 𝑒𝑥 + 𝑒−𝑥
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- 94. Differential Equations For theequation, 𝑑𝑦 𝑑𝑥 + 7𝑥2𝑦 = 0, if y(0)= 3 7 , then the value of y(1) is (a) 3 7 𝑒− 7 3 (b) 7 3 𝑒− 7 3 (c) 3 7 𝑒− 3 7 (d) 7 3 𝑒− 3 7
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- 96. Differential Equations The differentialequation 𝑑𝑦 𝑑𝑥 + 4𝑦 = 5 is valid in the domain 0 ≤ 𝑥 ≤ 1 with y(0) =2.25. Then solution of differential equation is (a) 𝑦 = 𝑒−4𝑥 + 1.25 (b)𝑦 = 𝑒−4𝑥 + 5 (c) 𝑦 = 𝑒4𝑥 + 5 (d)𝑦 = 𝑒4𝑥 + 1.25
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- 98. Differential Equations A differentialequation is given as 𝑥2 𝑑2𝑦 𝑑𝑥2 − 2𝑥 𝑑𝑦 𝑑𝑥 + 2𝑦 = 4 The solution of differential equation in terms of arbitrary constant 𝐶1 𝑎𝑛𝑑 𝐶2 is (a) 𝑦 = 𝐶1 𝑥2 + 𝐶2x +2 (b)𝑦 = 𝐶1𝑥2 + 𝐶2x +4 (c) 𝑦 = 𝐶1𝑥2 + 𝐶2x +2 (d)𝑦 = 𝐶1 𝑥2 + 𝐶2x +4
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- 101. Differential Equations Join MyTelegram Group https://t.me/dheeraj_sardana ✓ For daily dose to improve concepts ✓ For daily important questions ✓ For latest information about exams ✓ For notification of the class ✓ For important PDFs & notes ✓ For clearing the doubts by experts DHEERAJ SARDANA
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