10 Differential Equations Test Form A 1. Find the general solution to the first order differential equation: y 1 yy 0. 1 (a) (b) ln y 1 y ln y 1 C y y C y 1 C y 1 y C. Find the general solution to the first order differential equation: y d y 0. (a) (b) ln y y ln y y C C Cy y ln y C 3. In 1980 the population of a town was 1,000 and in 1990 it was 0,000. Assuming the population decreases continuously at a rate proportional to the eisting population, estimate the population in the year 010. (a) 17,619 (b) 18,000 19,048 18,141 4. A radioactive element has a half-life of 50 days. What percentage of the original sample is left after 60 days? (a) 43.53% (b) 49.56% 37.50% 5.00% y sin 5. Find the particular solution to given the general solution y C cos and the initial condition y 1. (a) cos (b) cos 1 cos 1 cos 6. Find the orthogonal trajectories for the family of curves y C 3. (a) y 3 K (b) 3y K 3ky 0 3y 7. The slope field for a differential equation is shown. Choose the equation that could be a particular solution to that differential equation. (a) y sin y (e) y ln (b) y 1 y e 3 3 3 3 y
Test Bank 103 8. Consider the differential equation with the initial condition y 0. Use Euler s method with h 0.01 to approimate y 0.03. (a) (b) 1.458 1.940598 1.940891 9. Find the general solution to the first-order differential equation tan y cos. d (a) y sec C (b) y ln y sec sin C ln y cos sin C y cos Ce sin L 10. The logistics differential equation produces y Find the value of b for the logistics 1 be kt. differential equation dp dt y y dt ky 1 y L 3P P given the initial condition 0, 15. 0 100 (a) 15 (b) 80 1 11 (e) 79
104 Differential Equations Test Form B 1. Find the general solution to the first order differential equation: 4 y d 0. (a) y C 4 (b) 4y y y C y 4 C y 4 4 C. Find the general solution to the first order differential equation: y 3 y 3 d 0. (a) 3 4 8y 3 C (b) y 3 ln C y 3 3 y 3 ln 3 C y 3 3 3 ln C 3. A certain type of bacteria increases continuously at a rate proportional to the number present. If there are 500 present at a given time and 1000 present hours later, how many will there be 5 hours from the initial time given? (a) 1750 (b) 88 3000 143 4. Choose the differential equation that matches the solution curves sketched in the slope field. 4 y (a) (e) y 1 y y y (b) y ln y 4 4 5. Find the particular solution to given the general solution y sin A B and the initial conditions y 0, y. (a) sin 1 (b) sin sin 1 sin 1 6. Find the solution to the initial value problem e y e y with the initial condition y 0 0. (a) (b) y ln e 1 y ln e 1 y ln 1 e y sin y ln e 1
Test Bank 105 7. Find the general solution of the differential equation: 1 tan y. d C (a) y (b) y C 1 1 cos y C 1 cos y C 1 8. Consider the differential equation with the initial condition y 0 1. Use Euler s method with h 0.1 to approimate y 0.3. (a) 1 (b) 0.956 0.198 0.970 y y 9. Find the general solution to the first-order differential equation 1. d (a) y C 3 (b) y ln C 6y C y y ln 4 C 10. Solve the Bernoulli equation y y y5. (a) y Ce 1 4 (b) y Ce 1 4 y Ce 1 y 1 Ce 1
106 Differential Equations Test Form C 1. Find the particular solution to the differential equation given the general solution y Ce 3 and the initial condition y 1 0. (a) y 0e 3 3 (b) y 0e 3 y 0e y 0e y 3y. Find the general solution of the differential equation y y 0. (a) y ln C (b) y y C y C y C 3. Find the orthogonal trajectories for the family of curves y C 0. (a) (b) y ln y3 3 ln K Ky 4y K ln 0 y ln y K 4. A radioactive element has a half-life of 40 days. What percentage of the original sample is left after 48 days? (a) 49.56% (b) 43.53% 5.00% 37.50% 5. Determine which function is a solution to the differential equation 1 (a) (b) ln y e y y 0. 6. Determine whether the function f, y 3 y 4y y 3 is homogeneous, and if so, determine its degree. (a) Homogeneous; degree 1 (b) Homogeneous; degree Homogeneous; degree 3 Not homogeneous L 7. The logistics differential equation produces y Find the value of b for the logistics 1 be kt. dt ky 1 y L dp differential equation given the initial condition 0, 15. dt 7 P P 10 300 (a) 14 (b) 99 13 98
Test Bank 107 8. Consider the differential equation with the initial condition y 1 1. Use Euler s Method with h 0.1 to approimate y 1.3. y y (a) 1.485714 (b) 1.3700841 1 1.1 9. Find the particular solution of the differential equation y y 4, y 0 4. (a) y 4 (b) y e y e y e 10. Solve the Bernoulli equation (a) (b) y y 1 1 1 1 Ce Ce y 1 Ce 1 y y y 3. y 1 Ce 1
108 Differential Equations Test Form D 1. Find the general solution of the differential equation: cos y tan y 0. d. Find the general solution of the differential equation: y d y 0. 3. A certain type of bacteria increases continuously at a rate proportional to the number present. If there are 500 present at a given time and 1000 present hours later, how many hours (from the initial given time) will it take for the number of bacteria to be 500? Round your answer to decimal places. 4. Sketch a slope field for the differential equation at the points indicated on the graph below. d y y 3 1 1 1 3 1 5. Verify that the equation y Ce is a solution to the differential equation y y 0. 6. Find the orthogonal trajectories of the family y C and sketch several members of each family. 7. Find the solution to the initial value problem: e y cos y y, y 1 0. 8. Consider the differential equation with the initial condition y 1 0. Use Euler s Method y y 1 with h 0.01 to approimate y 1.03. 9. Find the general solution to the first-order differential equation y y. 10. Find the particular solution of the linear differential equation y y sin sin, y.
Test Bank 109 Test Form E 1. Solve the differential equation: y y.. Find the general solution of the differential equation: 3. Find the particular solution of the differential equation 500 y that satisfies the initial d condition y 0 7. y e y. 4. The rate of change of y with respect to is inversely proportional to the square root of y. a. Write a differential equation for the given statement. b. Solve the differential equation in part a. L 5. The logistics differential equation produces y Find the logistics equation that for 1 be kt. dp dt 3P 5 dt ky 1 y L P that satisfies the initial conditions 0, 5. 350 6. Find the orthogonal trajectories of the family 4y C and sketch several members of each family. 7. Solve the homogenous differential equation: y y d 0. 8. Consider the differential equation with the initial condition y 1 1. Use Euler s Method y 1 y with h 0.1 to approimate y 1.3. 9. Solve, by any appropriate method, the first-order differential equation y y tan tan. 10. Find the particular solution to the first-order linear differential equation y y 1 with the initial condition y 1 1 5.