Mathematics-1 Tutorial 4 PDF

Parul University Faculty of Engineering & Technology Department of Applied Sciences and Humanities 1st Year B. Tech. Programme (All...

Parul University Faculty of Engineering & Technology Department of Applied Sciences and Humanities 1st Year B. Tech. Programme (All Branches) Mathematics – 1 (303191101) First Order Differential Equations Tutorial- 4 Q1. Find the order and degree of the following differential equations: 𝑑𝑦 2 𝑑2 𝑦 𝑑𝑦 2 a. ( ) + 2𝑦 = 𝑥 b. ( 2 )+( ) +𝑦=0 𝑑𝑥 𝑑𝑥 𝑑𝑥 2 3 𝑑2 𝑦 𝑑𝑦 2 4 c. (𝑑𝑥2 ) + + 3𝑦 = 0 𝑑2 𝑦 𝑑2 𝑦 𝑑𝑥 d. [1 + (𝑑𝑥2 ) ] = 𝑑𝑥2 2 𝑑4 𝑦 𝑑 3 𝑦 𝑑𝑦 𝑑𝑦 4 4 6 e. (𝑑𝑥4 ) + 𝑑𝑥 3 + 𝑥 3 (𝑑𝑥 ) = 0  dy   d2y  𝑑𝑥 f.   +4 = 2   dx   dx  Q2. Form the differential equation from the following: a. 𝑦 = 𝐴𝑒 2𝑥 + 𝐵𝑒 5𝑥 ; where A and B are arbitrary constants. b. 𝑦 = 𝑒 𝑥 (𝐴𝑐𝑜𝑠𝑥 + 𝑏𝑠𝑖𝑛𝑥); A and B are arbitrary constants. c. Family of circles of radius r whose centers lies on the x-axis. Q3. Verify that the given function is a solution of the corresponding given differential equation, where, a, b, c are arbitrary constants. 𝑎. 𝑦 ′ + 𝑦 = 𝑥 2 − 2, 𝑦 = 𝑐𝑒 −𝑥 + 𝑥 2 − 2𝑥 ′ b. 𝑥 + 𝑦𝑦 = 0, 𝑥2 + 𝑦2 = 1 ( ) Q4. Check whether the differential equation x 2 + y 2 + 3 dx − 2 xydy = 0 is exact or not? Q5. Solve the following Differential equations: 1. 2 xy dx + x 2 dy = 0 ( ) ( 10. 1 + y 2 dx = tan −1 y − x dy ) 2. xy '+ y = 0, y ( 2 ) = −2 dy 2 11. xy − = y 3e − x dx ( ) 3. x 2 + y 2 + 3 dx − 2 xydy = 0 12. xy (1 + xy 2 ) dy =1 dx ( ) ( 4. y 4 + 2 y dx + xy 3 + 2 y 4 − 4 x dy = 0 ) 13. dy = e x− y ( e x − e y ) dx ( ) ( ) 5. x 2 y − 2 xy 2 dx − x3 − 3x 2 y dy = 0 14. 2 xydx + x 2 dy = 0 6. ( xy + 2 x y ) ydx + ( xy − x y ) xdy = 0 2 2 2 2 15. dy − y = e2 x dx dy dy x 7. x + (1 + x ) y = x 3 16. +y=− dx dx y ( ) 8. x (1 − 2 y ) dx − x 2 + 1 dy = 0 with y ( 2 ) = 1 dy 17. 2 xy = y 2 − x 2 dx 9. ( 2 x − 4 y + 5 ) dy + x − 2y + 3 = 0 ( ) ( 18. x + 3xy 2 dx + 3x 2 y + y 3 dy = 0 3 ) dx Q6. The tank contains 1000 gal of water in which 200 lb of salt are dissolved. Fifty gallons of brine, each containing (1 + cos t ) lb of dissolved salt, run into the tank per minute. The mixture, kept uniform by stirring, runs out at the same rate. Find the amount of salt y ( t ) in the tank at any time t.