Question

(1 point) Given the third order homogeneous constant coefficient equation y" + 3y" + 3y + y = 0 1) the auxiliary equation is ar3 + br2 + cr + d = ^3+3r^2+3r+1 - = 0. 2) The roots of the auxiliary equation are -1,-1,-1 (enter answers as a comma separated list). 3) A fundamental set of solutions is e^(-x),xe^(-x),x^2e^(-x) (enter answers as a comma separated list). 4) Given the initial conditions y(0) = -1, ý (0) = 2 and y" (0) = -1 find the unique solution to the IVP y = -1e^(-x)+3xe^(-x)+3x^2e^(-x)

Answer 1

y" + 39" + 3y + y = 0 m² +30m² +3m + 10 (m+ 13 = 0 m =-1,-1,-1 Three equal roots General Solution is y(x) = cente + Gaxen + Człemse = y(x) = C, et + co xe* + C₂ 2 ² 2 Given y(o)=-1 y(0)=2 y"(0) Y(o)=c, toto -1 =C, (, = -1 9' () = (*)-) + Cible*%-1) + € *()) + z (x²(x-²)(-1) + é y(o)= (-1)(-) + C2(1) + C3(0+0) 2 = 1+to y"(1) = Ciést syl-(de*(-1) + 2) +6e)) +63f-ficcu) te%2x) + 2[8*0+ * (2+269)) Y"to) = (1 + C2 (-1-1) +63(2011) = -1 @ - 212 + 2C3 -1 - 1-2 +263 4 = 1