Question

Suppose we want to solve and plot the solution to the second order equation y'(x) + 8y'(x) + 2y(x) = cos(x); y(0) = 0, y'(0) = 1

Answer 1

(ر) y" () + 8y'(X) +zy lata easy ) FO y' (0121. y lola The auxiliary equation of the comesponding homogeneous equation is m²r omfa=0 m+ 8m + 16 = 14 >> (m+4)=14 ma-4. I VIY t Be . The funnet EMANTY). Be Eu Vur ution is Lemste complementary Let the particular internal be Yp - Y p = (eas a D²80&2 -1²+8D12 *** gorz (63) (eos x) = I-8D 1-64Dn (Cou I8D (cosa) 1-646-19) cosat 8 Sina 65 The general Solution (1) is y(x)= 04 (A FA A BETTER) + ega48 Sink 65 y (0)= (A +B) + ods Y"(T) = A e Euevia)? Gruevia) + Be (4-VM)? (4-VM) :P 80sn- Sina 65 y!() = A (-4+ NT) +B (-4-vrü) + 65 @ 1 -UCATB) + VTÝ (M-B) = 1- 8/15 »).-41-465) Arta (A-B) = 10.865

-> mu (A-B) 1- 4 65 65 ñ A 53 65174 East 5 ) 53 - Vlý 65VTM X2 => A= 53-V14 130 114 1 BE Go 53-rly 130VT4 -2V14 53 PVTY 130 VTY. Grý +53) 130VTY VT9X y (x) = e-uri e-uri [ 53-VTÝ 130VTY e Egthin xha :) e-ruz t Cosa of 8 since 65 salution lume.