Solve the differential equation. {image} 1. {image} 2. 3. 4. 5. 6. 7. 8. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

Find the solution of the differential equation {image} that satisfies the initial condition u(0) = 6. {image} 1. 2. 3. 4. 5. 6. 7. 8. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

Solve the initial-value problem: {image} y(0) = {image} . 1. 2. 3. 4. 5. 6. 7. 8. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

Find the orthogonal trajectories of the family of curves. {image} 1. {image} 2. 3. 4. 5. 6. 7. 8. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

Let y(t) and V(t) be the height and volume of water in a tank at time t. If water leaks through a hole with area a at the bottom of the tank, then Torricelli's Law says that {image} where g is the acceleration due to gravity. Find the height of the water at time t assuming the tank is full at time t = 0. Suppose the tank is cylindrical with height 5 ft and radius 2 ft, the hole is circular with radius 5 in and {image} . {image} 1. 2. 3. 4. 5. 6. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50