Ordinary Differential Equations & Vector Calculus PDF

ORDINARY DIFFERENTIAL EQUATIONS & VECTOR CALCULUS UNIT-I SHORT ANSWER QUESTIONS 1. Define exact differential equation. 2. Define the degree of differential equation with example 3. Solve 𝑦𝑑𝑥 − 𝑥𝑑𝑦 = 𝑎(𝑥 2 + 𝑦 2 )𝑑𝑥 4. Find the integrating factor of (𝑥 2 + 𝑦 2 )𝑑𝑥 − 2𝑥𝑦𝑑𝑦 = 0...

ORDINARY DIFFERENTIAL EQUATIONS & VECTOR CALCULUS UNIT-I SHORT ANSWER QUESTIONS 1. Define exact differential equation. 2. Define the degree of differential equation with example 3. Solve 𝑦𝑑𝑥 − 𝑥𝑑𝑦 = 𝑎(𝑥 2 + 𝑦 2 )𝑑𝑥 4. Find the integrating factor of (𝑥 2 + 𝑦 2 )𝑑𝑥 − 2𝑥𝑦𝑑𝑦 = 0 𝑑𝑦 5. Find the integrating factor of = 𝑒 2𝑥 + 𝑦 − 1 𝑑𝑥 6. Define orthogonal trajectory. 7. Find the orthogonal trajectory of 𝑥 2 − 𝑦 2 = 𝑎2 where ‘a’ is the parameter 8. Write the general form of linear differential equation in x and y terms 9. Write the general form of the Bernoulli’s Equation in ‘y’ and ‘x’ terms 10. Define self-orthogonal system of family of curves 11. State Newton’s law of cooling. 12. State law of natural growth. LONG ANSWER QUESTIONS 1. Solve (𝑥 2 𝑦 2 + 𝑥𝑦 + 1)𝑦𝑑𝑥 + (𝑥 2 𝑦 2 − 𝑥𝑦 + 1)𝑥𝑑𝑦=0 2. Solve 𝑦(𝑥𝑦 + 𝑒 𝑥 )𝑑𝑥 − 𝑒 𝑥 𝑑𝑦 = 0 3. Solve (1 + 𝑦 2 )𝑑𝑥 = (𝑡𝑎𝑛−1 𝑦 − 𝑥 )𝑑𝑦 𝑑𝑦 4. Solve (𝑥 + 2𝑦 3 ) = 𝑦. 𝑑𝑥 𝑑𝑦 5. Solve 𝑥 𝑙𝑜𝑔𝑥 + 𝑦 = 2𝑙𝑜𝑔𝑥 𝑑𝑥 𝑑𝑦 6. Solve + 𝑦𝑡𝑎𝑛𝑥 = 𝑦 2 𝑠𝑒𝑐𝑥 𝑑𝑥 𝑑𝑦 7. Solve (𝑥 2 𝑦 3 + 𝑥𝑦) = 1 𝑑𝑥 𝑥2 𝑦2 8. Find the orthogonal trajectory of the family of confocal conics 𝑎2 + 𝑎2 +𝜆 = 1, 𝑤ℎ𝑒𝑟𝑒 𝜆 𝑖𝑠 𝑡ℎ𝑒 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 9. Prove that the system of parabolas 𝑦 2 =4a(x+a) is self-orthogonal 10. Bacteria in a culture grows exponentially so that the initial number has doubled in three hours. How many times the initial number will be present after 9 hours 11. The temperature of the surrounding air is 20oC. The temperature of a hot body reduces from 100oC to 80oC in 10 minutes. What will be the temperature of the body after 20 Minutes? When will be the temperature 40oC? 12. The number N of bacteria in a culture grew at a rate proportional to N, the value of N was Initially 100 and increased to 332 in one hour. What was the value of N after 1½ hour? 13. Uranium disintegrates at a rate proportional to the amount present at any instant. If 𝑀1 𝑎𝑛𝑑 𝑀2 are grams of uranium that are present at times𝑇1 𝑎𝑛𝑑 𝑇2 respectively, find the half -life of uranium. 14. If 30% of radioactive substance disappears in 10 days, how long will it take for 90% of it to disappear. UNIT-II SHORT ANSWER QUESTIONS 1. Solve ( D 2  3D  4) y  0 2. Write the general solution of ( 𝐷3 − 𝐷)𝑦 = 0 3. Find the complete solution of (𝐷4 + 16)𝑦 = 0 4. Solve (𝐷2 + 2𝐷 + 1)𝑦 = 𝑒 −𝑥. 5. Find the P.I of (𝐷2 + 1)𝑦 = 𝑥 6. Find the P.I of (𝐷2 + 2)𝑦 = 𝑒 𝑥 𝑐𝑜𝑠𝑥 7. Solve (𝐷2 + 4)𝑦 = 𝑠𝑖𝑛2𝑥 8. Solve (𝑥 2 𝐷2 − 4𝑥𝐷 + 6)𝑦 = 0. 9. Define wronskian of two functions and give an example. 10.Write the general form of Euler- Cauchy’s linear equation of order LONG ANSWER QUESTIONS 1. Solve (𝐷3 + 2𝐷2 + 𝐷)𝑦 = 𝑒 2𝑥 + 𝑥 2 + 𝑠𝑖𝑛2𝑥 2. Solve (𝐷2 + 5𝐷 − 6)𝑦 = 𝑠𝑖𝑛4𝑥 𝑐𝑜𝑠𝑥 3. Solve (𝐷2 + 3𝐷 + 2)𝑦 = 𝑥𝑒 𝑥 𝑠𝑖𝑛𝑥 𝑑2 𝑦 𝑑𝑦 4. Solve the differential equation 𝑑𝑥 2 − 4 𝑑𝑥 + 4y = 8𝑒 2𝑥 sin 2𝑥. 5. Solve(𝐷2 − 1)𝑦 = 𝑥𝑠𝑖𝑛𝑥 6 Solve (𝐷2 + 4𝐷 + 3)𝑦 = 𝑒 𝑥 𝑐𝑜𝑠2𝑥 − 𝑠𝑖𝑛3𝑥 7. Solve (𝐷2 − 2𝐷 − 3)𝑦 = 𝑥 3 𝑒 −3𝑥 8. Solve by the method of variation of parameters(𝐷2 + 1)𝑦 = 𝑠𝑒𝑐x 9. Solve (𝐷2 + 4)𝑦 = 𝑡𝑎𝑛2𝑥 by variation of parameters. 𝑑2 𝑦 𝑑𝑦 10. Solve the differential equation 𝑥 2 𝑑𝑥 2 − 𝑥 𝑑𝑥 + y = logx 𝑑3 𝑦 𝑑2 𝑦 𝑑𝑦 11. Solve (𝑥 3 𝑑𝑥 3 + 3𝑥 2 𝑑𝑥 2 + 𝑥 𝑑𝑥 + 8) 𝑦 = 65 𝑐𝑜𝑠(𝑙𝑜𝑔𝑥) d2y dy 12. Solve ( x  1) 2 2  3( x  1)  4y  x2  x 1 dx dx UNIT-III SHORT ANSWER QUESTIONS 1. Define Laplace transformation, and 2. What is the existence condition of Laplace transform? 3. Define an exponential order function and give an example. 4. Find (i) L{𝑒 −2𝑡 (4𝑐𝑜𝑠3𝑡 + 𝑠𝑖𝑛2𝑡)} (ii) L{𝑡 𝑠𝑖𝑛𝑎𝑡} 𝑒 −𝑎𝑡 −𝑒 −𝑏𝑡 1−𝑐𝑜𝑠𝑡 𝑒 −2𝑡 𝑠𝑖𝑛3𝑡 5. Find (i) L{ } (ii) L{ } (iii) L{ }. 𝑡 𝑡 𝑡 6. Define periodic function and give an example. 7. State first shifting theorem of Laplace transform. 8. State convolution theorem. 9. Define Unit step function find its Laplace transform. 10. Dirac delta function and find its Laplace transform.  𝑠 11. Find 𝐿−1 {(𝑠 2+𝑎2 )2 } 2𝑠+12 12. Find 𝐿−1 { } (𝑠 2+6𝑠+13)2 LONG ANSWER QUESTIONS 1. Find the Laplace transform of 𝑓(𝑡) = {𝑐𝑜𝑠𝑡 𝑠𝑖𝑛𝑡 0