Question

1. Find the general solution for each of the following differential equations (10 points each): (a) (b) (e) y" – 2y! - 3y = 3e2x y" — y' – 2y = -2.3 + 4.2? y" + y’ – 67 = 1234 + 12e-2x y" – 2y' – 3y = 3.ce-1 y" + 2y' + y = 2e- (Hint: you'll use Rule 7. at least once) (e 2. Find the solution to the following differential equation with initial value (20 points) Y' + 2y' + 5y = 4e-* cos(2.0) with y(0) = 0 and y(0) = 1 To solve Problem 2, first find the general solutions as in Problem 1 and then use the initial condition to find the correct coefficients.

non-homo 2nd order linear equations

Answer 1

0 0 Given that y"-24-3y=3e2n The Auniliary equation of y"-24"-34 zo is fimizo => mi-2m-3 Q- E-uit wenk o= ({-u) it (E-usula (s (مقاما) (مناک 0 ) سات - غ 331 are the roots of frm=0 The complementory function 9ccmacie "tree Now Particles Integral Ipino a ㅗ DY_2D-3 بيا зе. = kan XE = uzat ezy 2_2x2-3 4-11-z y Pino - - een The general solution of y"_24134=3e? is y=ycem typing Yr cientizen-e y 29 6 Given that y"_y'-2y=-20 tun Ć 5-D-214 = un-an, The Aumiliary equation of con. -D^2ly 20 is mi-m-2=0 Dad -) frm120 > Ort-utusuk al mim-2) +1(m.27.20 >) (mal) (n=2) 20 =) mar!, 2 The complementory function Yenjacie tc2een particuler. Integral y pim), a I am 1 (UM-22) CO-D-2)

(un-an) -=[1-10-03 Croll s'1to+D+3... [-(05-09 j'emasan) Elit § 107-1) ++(94+9=203 1 + $506-305+3934 -D3)...] (um-2m) Ž[uñ-an tut (et u8-8) ) +8-0) Ť [un-zin +5-un +2 2 tulo +10)...] (ou Druno_21)=87-2 DY cun-20)=8 Sprm a 1 [uñ-on to The general solution is D3 lun-207=0 y = yorm typim ciente2e2n- 2n-1 cun-int) L flze DE an منہ Given that y "ty-ty=128 stizezn (8+D-6ly = 1203 ē zn The Auniliary equation is from=o mtm-t20 =) +3m-am-620 =) mimi3-21m+3)=0 a) (mtocm-2)=0.=) m =-3,2 The roots of fim720 are -3,2 The complementory function yerniacie 3"tchen particuler Integral yprm fro) = 12 * 1 i amo 1540-6 (12 e3" 412 e 2n ein Š toob ypcma *12* Front een 12 9+3-6 + 12 ean 4-26 it tot een ypinzani tena the te

Zein -3e2n Ypinia The genered solution is yinla yerni typim) "allymacie 3"the an 30 -27 +2e >Be d =D Given that y"_24-3y=inen =) (5-2D-319=onen dn The Auniliary equation i femi=0 = m-2m-320 ) un-3m+m-3=0 I memasti (m-3 =0 =) (mal)(n-3) 20 =) mar, ma3 The complementory function. Yerry a cie" the 3n. Particenter Integral Uprimo al arm) fco 3nen Now . - 3 en x [CD+I)-200+0.-3] ń lucro eam eanalu festas Esen n 0+1+30-70-2-3 = 30mm .se (m) ร) é -DY n Cinta I'm a -32"[mtoto ...] =) Spin) = -3nem The general solution is Y = 9crny typin). 4 374 9(n) = 8 +62 3 Bnen

Given that y"+zyltyazen =) (8+20+12y=zen The Auniliary equation is fimo ao > mtamtao a) cnt=0 mol, al. The complementory function Ycomo sere Caitlansen Now ypin) = I Qimo fCD) f(-1)=0 ( CW+20+1) 2x nen [ Po he ean 2D+2 nean franzo flranto - kx f'ldi z yermo -ñen f1a720, f"(o) flcalao, f\[2+0] The general solution is y = yern) typin) y = caitrange taven