M427J Exercises for Chapter 2 Section 2.1 1. Solve the IVP y + y 2y = 0, y(0) = 2, y (0) = 1. 2. Solve the IVP y + y 2y = 0, y(0) = 1, y (0) = 7. 3. Find the general solution of 2y + 3y 4y = 0. 4. Show that y1 = cos t and y2 = sin t are solutions to y + y = 0. Then show they are linearly independent. Then show that any linear combination of y1 and y2 is also a solution to the DE. 5. Show that y1 = 1/t and y2 = t4/3 are solutions to 3t2 y + 2ty 4y = 0. Then show they are linearly independent. Then show that any linear combination of y1 and y2 is also a solution to the DE. 6. If L = 2D2 + D 4I, nd the general solution of L[y] = 0. Section 2.2 Solve the following DE's and IVP's. 1. 6y 5y + y = 0, y(0) = 4, y (0) = 0 2. 4y y = 0, y(2) = 1, y (2) = 2 3. 4y y = 0, y(2) = 1, y (2) = 2 4. 9y + 9y 4y = 0 5. y + 6y + 13y = 0 6. y 2y + 5y = 0, y(/2) = 0, y (/2) = 2 7. 4y + 12y + 9y = 0 8. y + 4y + 4y = 0, y(1) = 2, y (1) = 1 9. 5t2 y 3ty + y = 0, t > 0 10. t2 y 3ty + 3y = 0, y(1) = 2, y (1) = 3 11. 5t2 y + 3ty + y = 0, t > 0 12. 4t2 y 8ty + y = 0, y(1) = 1, y (1) = 4 13. 9t2 y + 21ty + 4y = 0, t > 0 14. 4t2 y + 8ty + y = 0, y(1) = 7, y (1) = 3 15. t2 y t(t + 2)y + (t + 2)y = 0, t > 0. One solution is y1 = t. 16. (t 1)y ty + y = 0, t > 1. One solution is y1 = ex . Sections 2.3 and 2.4 Use the method of Variation of Parameters to solve the following DE's and IVP's. 1. y + y = tan t, 0 < t < /2 2. y 2y + y = et /(1 + t2 ) 3. t2 y 2y = 3t2 1, t > 0 4. t2 y + 4y + 2y = t2 , t > 0 5. 4y + y = 2 sec(t/2), < t < Section 2.5 Use the method of Undetermined Coecients (Judicious Guessing) to solve the following DE's and IVP's. 1. y 2y 3y = 3e2t 2. y 2y 3y = 3et 3. y + 2y = 3 + 4 sin 2t 4. y + 4y = 3 sin 2t, y(0) = 2, y (0) = 1 5. y + 2y + 5y = 4et cos 2t, y(0) = 1, y (0) = 0. Section 2.6 under construction