Question

Find a general solution to the given equation for t<0. y''(t) – Ły'(t) + 5 -y(t) = 0 t The general solution is y(t) = (Use parentheses to clearly denote the argument of each function.)

Answer 1

(1) Que - Find a general solution to the given equation for to. S (t) - 는 sitt) + (t) : + At) Co. Sueln "(t) - L 9't) + S Jt) 27 A ty" - ty' + 5% = 0 12 >) t-t+52 + 0 (0 2 let t=e2 = ㄸ logt. -2 New dy dt dy d2 dz dt A = 는 - 士 (3) . dy dz 분 Now БАР dt d df () () 는 분 + )

It's auxiliary equation is dz ; where =d -20 +5) Jeo C dz2 (4) 02 3Rs t 위은 Бар ㅑ dz dz dy hath d22 d2y from (1) 2P dz 22P бүP RP 키 여 KP 2. dt? dz БР Т RP 22P -Р dz - - 쿠쿠 루 (2)way} : () 뭐 () 뭐 - + = ( PT + P 은 은 t +² dz 읽은 PAP dz.dt FP2P MP 정P dtd t 귀분 t GP 하 2 2P P (2)

(3) m²_ m-2m +5 20 af ox?+bx+co by Shridharacharya formula by Shricharacharya 2 + 14-4x1x5 Formula a=-bz Jb²uac m = 29 2x) = gt 14-20 2 - 2 + -16 9 =2& Ji6i? 2 1 4i 9 2 Hence where A and B are m= Itzi m = 1+ 2i 1- zi Then the general salution of (41) is y = é CACE A COL 22 + B Sinzz] arbitrary Constants Then using (2) {t=e? and zz logt) ya t [A Cas 2 ( log t) + B Sin 2 (logt) y OR y(t) = t[ A cas (zlog t) + B Sin (zlogt)]

(4) OR 84)= t[ A cos (log t?) + B Sin Clog +2)] Which is the general solution of given differential equation •