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hw9_soln

# hw9_soln - MTH 2201 Differential Equations Fall 2009...

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Unformatted text preview: MTH 2201 Differential Equations Fall 2009 Homework 9 : Second Order Linear ODE ﬂ 1. In the problems below, y1(cc) is a solution of the given DE. Use reduc— tion of order method to ﬁnd the second solution of the DE. (D (a) y” - 4y, + 43! = 0; y; = 62‘”. (Solution: yg : 3362””) @(b) y” + 163/ = 0; y1 = 0084113. (Solution: y2 = cos 495) @(c) 9y” — 123/ + 4y = 0; y1 = 6237‘. (Solution: y2 = 936273.) @(d) 3223;” — 7my’ + 16y = 0; y1 = x4. (Solution: 3/2 = a34ln Incl.) @(e) (1 — 2:1: — 2:2)y” + 2(1 + \$)y’ — 2y 2 0; yl : a: + 1. (Solution: 3/2 = m2 + a: + 2.) 4—, 3 2. Find the general solution of / (D (2') 4y” + y’ = 0. ® (ii) y” + 8y’ + 16y = 0. (ii) 8y” + 23/ — y = 0. (iv) 3;” — y = 0 3. Consider y” + y' — 6y = 0. (i) Compute the solution ¢ satisfying 45(0) = 1, (15’ (0) = 0. (ii) Compute the solution 10 satisfying 7M0) = 0, ¢’(0) = 1. .4. Find all solutions qt of y" + y = O satisfying ¢(0) : 1, ¢(7r / 2) = 2 5. Let qb be a solution of the equation y” + aly’ + (12:9 2 0, where a1, (12 are constants. If 1/)(75) = e(”1/2)t¢(t). Show that 1b satisﬁes the DE 31” + kg = 0, where k is some constant. 6. Determine the values of a, for which all solutions of y” — (2a — 1)y’ + a(a - 1);; = 0, tend to zero as t —> 0. 7. Find the general solution of d5y dy - III _ 4 II __ I Z '- _ Z @(2) 2/ d4 y 5:2 0 (2%) M5 16 d9; 0 y 1/ @(22’6) 1685;; + 24rd}; + 9y 2 0. /€ﬂd(l. 4. %//+3:0 Y: i; ‘3“): c, 005x + cg, sinx ‘j/(X): ~ ‘ CI <1m>< -l— 6:. 08% ﬁx} :- @0376 + a sinx C ” ’ ‘3 +aly+4zazc> k:&_ W L _ ”‘4' m: 2 ( 4t/z)+ aw &* ‘V 6‘ M) - a a” VI ,8 ( +£~Jﬁo ?CX): @ Hf “— [e + C2 6J7“: why/{\lr—i’fe _. a, ”\$542. 2. of: — “(t Z”:* C! e + C2. ecy-,;;é {-70 (36*) : «M: K”? 7. (i) 3”’_4-7”——57’= 0 r3 ~<£V7'—g'r =o For—9mm) = o . 01" d "’ ’0??? ' (4227?» (9’05) '3 C: + CzCD'SZK + C3 S‘D’IZX + Cq. €ZX+ Cfe—ZX 001—0603ij mi?— 12 1 + CL\$ 1X+ngeﬁsﬁx+C¢KSMFK 5 2. 7:: ...
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