Homework12

Section 4.2 13.5, 10, 13,27,34) 3.y " + 5y' 6y = 0 anx equation: r+ 5r+6 0 => roots: r, = 2, r= - 3 It basis of solutions: y, e, yz e + general solution: y cie 4 ce- t, a.CzER 5. y " + 8y' + 16 y = 0 anx equation: r8r+16 0 => roots: r, = - 4, rz - 4 4t basis of solutions: y, e4*,yz e general solution: y ce 4* xe 4,a.CzER 10. Y" y'-ly 0 1 + 355 anx equation: r-r-11 0 => roots: r, = 2,(z 1-355 11 + 355/2) t 11 355/2) t basis of solutions: y, e · yz e 11 355/2) t general solution: y c, e11+ 355/2) + + cze 13. y " + zy'-8y 0, y 10) 3, y',0) =-12 anx equation: r+2r-8 0 => roots: r, 2, rz = 4 basis of solutions: y, e2+, yz e 4t general solution: y cie + xe 44, a.CzER 3: TS.]= [2] y2= I5 ? 7:[-x] [e-xi22] a = 0. 3 => y 3e 4t 27. Since sin(2+) = Isint cost => y z zy, xy, and ye are linearly dependent 34. (a) Y, It) yelt) y, (t) y2lt) y, (t)yi(t) WTy,yz]t y,it) y': It) 13 ypnsT7=707 ifde+=o=> it is invertible and columns are linearly independent => y, (t) and yet) are linearly independent (2) If y, ct) and yet) are linearly dependent, it is not invertible => det = 0 => WTy,, yz1 (t) 0

section 4.3 11,3,21,24,29b) 1.y " + 9y 0 anx equation: 4 9 = 0 => roots: r, 3i, r2 - 3i 12 0, = 3) basis of solutions: y, cos(3t ), yz sin13+ general solution: y C,cos(3t) + (2 sin13t), C.CzER 3. E "- 6E'+10E 0 anx equation: r26r+10 = 0 => roots: r, 3+1, r2 3 -i (2 3,B 1) basis of solutions: y,= est cost, yz= etsint general solution: y cecost+ (zesint,C.CzER 21. y" + 2y' + 2y 0, y 10) 2, y'(0) 1 anx equation: r+20+2 0 => roots: r, = -1+1, rz = -1 i (6 -1,B 1) basis of solutions: y, etcost, yz= etsint 31 TY.] [i]. ": I, 77,] I, i ? ] C1 2, Cz = 1 => solution: y zetcost esint 24. y" + 9y 0 yc0) 1, y'c0) = 1 anx equation: 12 + 9 = 0 roots: r 35, rz - 3i (6 0, = 3) basis of solutions: y, cos(3t ), yz sin13+ 31 T, s7 8]. "Ty773] [8s; ! ] a = 1, a s general solution: y c0S(3t) + 113 sin13t> 29b. y'" + 2y" 5y' 26y 0 anx equation: 1202 + 50-26 0 => roots: r, 2, r2 = 2 3i,r3 = 2 - 3i(2 = 2,B 3) basis of solutions: y1= e,yz etsinit), y3 e*cos(3+) general solution: y ciet+ce-sin1st) + cecos13t), a,22,3 ER