Question

Given that yı(t) =ť is a solution to the ODE: ty' + 4ty – 10y = 0, use the reduction of order method to find another solution y2 corresponding to the initival values ya(1)=- and y(1)= y2(t) =

Answer 1

solution: consider, t²y" +444404=o f Yeld) = t2 .. i say Yalt= yeld) VIA) a taviti Now let =) Ya'lf I = PA VIJI + t²o'la) & yllf) Iavit) + 2 viit) + 2a VIA). (flid att Iflat (flipfot (fliff = 1. a Put these intolis, tt v"H) + 4730114) + 24. 2014) +8f20(Alt at301#) -105264) 0 lie, 44191174) +8+381(t) = 0 Now let w=0' WW w oll 80 x4 wit&f3 w=0 - W't B.w=o = C =) WiFlie = ler xodt + 6 =) Wet). 2800et But = WCH)=67-8 je, g1+) = swid) dt + k + s (4-8d4+4

# $it= *** --744 80.4260) = 42 916) = -(4+5+642 *43!10) = +9c4-6+2kt.! Now using initial conditions we have - = 4214) = - + K -1=-C+7k -las Y2'13) = 5C +2k 7 ) 5 = SC +14 K 16) Now Now by (6) + 5819); 5 = 54 +141 -*= $c +35K O = 49 K - 49 K lie, CE I using (ay.com Hence, | 426+) = -57-5