Question

Question 5

5. Find the general solution using variation of parameters Y" - y'- 2y 2. an -t

Answer 1

Given differential equation y"- y'=2y = zet consider Homogenous differentied egn y"_y!-3430 Auxilludy equation is m²m-220 m-am tm-2-0 mim-2)+1(m-2920 (M+1)(m-2)=0 m+120, m-220 maal, mar Roots are different mil, 2 complementary solution 3 4cc cette 2 t

now to find particular solution Up = 0,4, + V2Y2 Suppose 4,2 et Y₂ = et 8(4) are zë "lanet 42 = 2e2t To save the system of equation vily, tUaY2 20 vily,'t Val4lar(A) 4 2 vicēt) tus! (et)=0 vikét) +va (net)=é 3 27 4 Adding eq" & eq" 6

viett vele?t=0 - viet tavaltaret 3 un'e ?t=zet Vala zet 3627 1) WIN e -34 volebes inteqreeting 3 festas V2 = 2 Vza -3t WIN Nga -Bt e eyxaney zviet tavalto -viet tavalt net + arlet - - 24 Vila t درر ) 01002 + vi's - 2 2

integrating on on both sides " -2 t 3 Particular solution i Up = vyt VzY2 a terch (72 -37 2t e Че - 2 * e 3 "Z tét general solution is, 9 . Y = Yet up [yeas: Giöttöpuiki 9