Question

solve the initial value problems:

a) y'' + 14y + 49 = 0, y(0) = -1, y'(0) = 0

b) y'' + y - 2 = 2sin(x), y(0) = 1, y'(0) = 2

c) y'' - 10y' + 25y = e^(5x), y(0) = 1, y'(0) = 6

Answer 1

@ y144 +19=0 Constants. y = - 49 is a linear different an equation with constant coeffacients. xiliary equation for 24-4 198=0 is m2 +19=0 ~ me fuqi complementary for function is G COB (VTq x) + q sin (Van) where a, le are arbitrary To find particular solution, I use Method of variastons of parameter, Let the particular solution be 9 (1) = y() Col(1179) +ų (a) Sh(viau) Then yw = 2 440(172) + V's!(W7 ") } +2- VTG v sin (Vign) + VTq Vy los (Tan)

We choose V putan e v v such that (VT) + V,' sin (VT) = * -va y sin (1750) Va y sin cvia ») + VT4 v_ cos(vaw) 0p (*) 3-va visin (V1w) -4 vya vil dog (wwws +2 -14%, cos (Van) – 19V, sin (WIM) sitting these in the given differentia putting thesk in the equation we have We IC) +19%(K) = - 19 - V75 y’sin (172 v) 4 Via 15! Cob( VTG MO) = -49 > - 'Sin (VT ) + 1;!cos(174) --- — soeving 0 f, we get sin(1796) 3v; loj[1792) 4 v," pin (vian) + 80 ( 1752) 3-v,' s'u (178) + V5! Cos (1749) Sv - ha cai (17x) 609 (174 ) a - vel sir(71)+ clas" (Van) Y = -4V V = 7 cos (VTq x)

From ② , pusing Vil we get y' cas (1793) - Cay (Vran) 850 (Via) - 0 -> V = 7 7 in (VIM) V 27 V 2 sin (VIM) => y()=-V. Cox (UTAW) -- (van) V14 V!(9) -71 65(Utax) > 1, --712. Sin (VTow) - - 7 sin(vaw) 7 V = -7/7. VIA - % (2)= - Z cas" (179) – Z sin (Vigo) de Grenerar solution for yr + 197 +49 50 is YOU)= G cos(174 a) + G sin(utan) - vow, y(n)= - V79 C, sih( vta ") + VTA 6, 43 (Van) y laat gives G-2/2 = - + 7 6 - ]

710=0 gives Vig ( 20 = Cq = 0 The required solution you car (vīga) -