Question

2) (Higher-order Equations) Find a general solution to the equation t” +tv – 2t + 2y = sint, t> 0.

Answer 1

t>O Z det t =e . Z 3 ty" + 1"y" - 2ty'+ 2y = t3 sint = dt = e? es de 4 da e dy de l = t de de di da tdy = dy dz at dy L dy da + 2 042 t . dar de t dry - 1 de ab + db Ł 2 ) dey 2² d df 2 dy dz dz2 Y di dy de3 = + a²y dz t2 dal dt 2 day Es dz2 +420 t2d23021 17 de 1 a y d y = a d'y (3 d23 2 3d3y df 3 dy dz 3 dly - d by da3 + > dz? dy -2( 22 )+24 .. (2 ) + ( ) - 2 ()+2% 3 Z е sih (@2) N ） de med en ene e per zy = ? an(e?) 3 Auxiliary equation corresponding homes geneous equation 3 mt2 2m2 m -O =) m2(m-2) - 1/m-2)=0 » m -2) (mt)(n+1) = 0 m =-1, 1, 2

The Complementary function e, e2 27 у 2 at e } Egt The particular integral १६ 22448 (B-2 -2) (0-1) (D+1) 32 Sin ez 7 a a ī Biz (e? sin e2) L 2 0 * 5-2-4 57+5+) Fas (e?? In ez) the t (?? que?) = $. e o CEPS - I. ?tle"? Sime?] t le Sin e²] z 4Z + 16 ez 22 e sin e² h e : wr apavus za 2} - 20 za uspens zo za ug 22 l e . + za & نی (- Sint t L t ( +6 El cosi)-6.(Siny *cost + + 2 e 22 = 3 12 / ene de ut t co sin edt + 4 4 jenint de 1/8 t? {Cost) - t. (z. (- cond) fint)) t² {Cost) -3th Esing $2%. Cost + b + cost - 1 tsent - Cost - Sent Thus the complete solution y(f) = G ħ t t t e ť Itsenttlost sint 11 + = 7 2 Sent + Cost ㅋ G E, G are Conftants.