Question

Solve each of the following.

7. Solve each of the following: (a) y" - 3y' + 2y = 0 (d) y" - 4y' + 5y = 0 (b) y" - 10y' + 25y = 0. (C) y" + 3y' - 5y = 0 (e) y" + y = 0 subject to y(1/3) = 0, y' (1/3) = 2

Answer 1

a ''-34'tayao consider auxillon egon m² - 3 mta-o m²-am-mtazo mm-2)-ilm-2)=0 (m-)) (m-2) 20 → mH com- do Malf m=2 noots are distiche 4(X) = ciemix 20m₂ x (0) - (iedt een yllt loy'+25y = 0 auxillars an m²-lom +25=0 (m-5) 2 - 0 m=sos - rofs are repeatedl i 4 (M) = cremat caren. z (cit (2x) emn Y(n) = ((1+(2n) est. y + gy'-sy=0 auxillam el m² tom-s-o 1

m² +3m-5=0 - - 3 + vg+20 m = -3 + 129 m 2 mis - 3#vag me -3-12 & YM) - ciemint ca e man - 3+ 129 x ya) = le -3 -128 x & le 2 yll – 4 4 4 sy=0 Ouxillary en m2-4m + 5 = 0 - - (-4) I 16-20 4 + V-G +2 2 - -27 m=2 t t = atbi a= 26=1 1ъ ) оте со ур Телан бо Y(A) = & ad (cicosbx+lasinda) 7 y) = e 2" (cicosx & casinx) 1

@ yllty=0 auxillon eon m²+1=0 m = 19 a=096=1 roots are complex y(x) = x(cleosa+(asina) 4x) = cicosat (a sinn I 7 0 1 cicos It casin I = 0 Ao Y'(a) = acisinn & acom. y (17/3) = 2 -ci son 3+ (2 Corrla = 2 - (1 6 + c2 t = 2 2

muliply oy b & 6 by the then add (2 + 2) 2 = 2 2 2 s (2=2 2 hm 6 cita = -(2 82 il- – 43 SV2 V 2 - U Yin) - - 43 coox of 4 12 sinn