DIFFERENTIAL EQUATIONS

1. Given (2y^3 – x^3) dx + 3xy^2 dy = 0; when x=1, y=1. Find y when x=3

a. 7.16 c. 6.17b. 1.76 d. 5.16

2. Given (y^4 – 2xy) dx + 3x^2 dy = 0; when x=2, y=1. Find y when x=3.

a. 6.12 c. 1.22b. 5.22 d. 3.64

3. (x^2 + 6y^2) dx – 4xy dy = 0. Find the solution when x=1 , y=1.

a. 2x^2 = y^2 (3x-1)b. y^2 = 2x^2 (3x+1)c. 2y^2 = x^2 (3x-1)d. x^2 = 2y^2 (3x+1)

4. Find the solution of (x^4 – 4x^2y^2 – y^4) dx + 4yx^3 dy = 0. When x=1, y=2.

a. x^2 (5-3x) = y^2 (5+3x)b. y^2 (5+3x) = 2x^2 (5-3x)c. x^2 (5+3x) = 2y^2 (5-3x)d. y^2 (5-3x) = x^2 (5+3x)

5. Find the solution of xydx – (x^2 + 2y^2)dy = 0a. y^2 = 4x^2 ln|y/c|b. x^2 = 2y^2 ln|x/c|c. x^2 = 4y^2 ln|y/c|d. y^2 = 4x^2 ln|x/c|

6. Find the solution of (x – 2) dx + 4(x+y-1) dy = 0. a. 2(x+1) = -(y+2x) ln (c|x+2y|)b. 2(y+1) = -(x+2y) ln (c|x+2y|)c. 2(y+1) = (x-2y) ln (c|x+2y|)d. 2(y-1) = -(x-2y) ln (c|x-2y|)

7. When a bullet is fired into a sand bag, its retardation is assumed equal to the square root of its velocity on entering. For how long, will it travel if its velocity on entering the bank is 144ft/sec.

a. 24 secs c. 32 secsb. 10 secs d. 12 secs

8. An object falls from rest in a medium offering resistance. The velocity of object before it reaches the ground is given by dv/dt + v/10 = 32. What is the velocity of the object one second after it falls?

a. 40.54 c. 30.45 b. 38.75 d. 36.78

9. A tank initially holds 100 gal of brine solution containing 20 lb of salt. At t=0, fresh water is poured into tank at the rate of 5 gal/min, while the well-stirred mixture leaves the tank at the same rate. Find the amount of salt after 10 mins.

a. 13.12 c. 12.13b. 11.23 d. 15.11

10. A tank contains 200 gal of salt solution in which 100 lbs of salt. A pipe fills the tank with salt solution at rate of 5 gal/min containing 4 lbs of dissolved salt. Assume that the mixture in the tank is kept uniform by stirring. A drain pipe removes the mixture from the tank at rate of 4 gallons/min. Determine the amount of salt in the tank after 30 mins.

a. 300 c. 420b. 520 d. 600

11. A tank initially contains 200 liters of fresh water. Brine containing 2 kg/liter salt enters the tank at a rate of

4liters/min and mixture runs out at 3 liters/min while being kept uniform. Find the amount of salt after 30 mins.

a. 227 kg c. 300 kgb. 158 kg d. 197 kg

12. Given (D^2 + 4D + 4)y = 0; when x=0, y=1, y’=1. Find y when x=1

a. 7.23 c. 5.32b. 2.07 d. 0.27

13. Given (D^2 + 4D + 4)y = 0; when x=0, y=1, y’=1. Find the particular solution.

a. y = (1-x)e^-2xb. y = (1+x)e^2xc. y = (1+x)e^-2xd. y = (1-x)e^2x

14. Given (4D^2 – 4D + 1)y = 0; when x=0, y=-2, y’=2. Find y when x=2

a. 4e c. 8eb. 6e d. 10e

15. x’’ + 4x’ + 5x = 8sint; when t=0, x=0, x’=0. Find the particular solution

a. x = (1+e^-2t)sint + (1-e^-2t)costb. x = (1+e^2t)sint – (1-e^2t)costc. x = (1+e^-2t)sint – (1-e^-2t)costd. x = (1+e^2t)sint +(1-e^2t)cost

16. Find the particular solution for (D^2+1)y=10e^2x; when x=0, y=0, y’=0

a. y = 2(e^-2x – cosx – 2sinx)b. y = 2(e^2x – cosx – 2sinx)c. y = 2(e^2x + cosx + 2sinx)d. y = 2(e^-2x + cosx – 2sinx)

17. Given x’’(t) + 4x’(t) + 5x(t) = 10; when x(0) = 0, x’(0)=0. Find the particular solution

a. x = 2(1 – e^-2t cost – 2e^-2t sint)b. x = 2(1 + e^-2t cost + 2e^-2t sint)c. x = 2(1 – e^2t cost – 2e^2t sint)d. x = 2(1 – e^2t cost + 2e^-2t sint)

18. y’’ – 4y’ + 3y = 2cosx + 4sinx. Find the general solution.a. y = C1e^x + C2e^3x - cosx b. y = C1e^-x + C2e^-3x + cosx c. y = C1e^-x + C2e^-3x - cosx d. y = C1e^x + C2e^3x + cosx

19. 4y’’ + y = 2; at x=, y=0, y’=1. Find y when x=2.a. -0.77 c. -5.67b. 0.30 d. 3.77

20. 2y’’ – 5y’ – 3y = -9x^2 – 1; at x=0, y=1, y’=0. Find y when x=2

a. 5.64 c. 9.35b. 6.54 d. 12.85

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