MA 36600 LECTURE NOTES: FRIDAY, FEBRUARY 273Then the general solution to the nonhomogeneous equation is the functiony(t) =c1y1(t) +c2y2(t) +Y(t)for some constantsc1andc2. We remark that typically findingy1andy2is easy: when the coeﬃcientsa(t),b(t), andc(t) are constant functions, one can computey1andy2by considering roots of the characteristicequationa r2+b r+c= 0. We will spend the next couple of lectures discussing how to find a solutionY.Note that onceonesolutionYis found, we can computeallsolutionsy.We give a proof of the claim above. Say thaty=y(t) is any solution to the nonhomogeneous equation,and letY=Y(t) be as in assumption (iii). Denote the differenceu(t) =y(t)−Y(t).We see thatu=u(t) is a solution to the homogeneous equation:a(t)u+b(t)u+c(t)u=a(t)y−Y+b(t)y−Y+c(t)y−Y=a(t)y+b(t)y+c(t)y−a(t)Y+b(t)Y+c(t)Y=f(t)−f(t)= 0.By assumption (i), bothy1andy2are also solutions toa(t)u+b(t)u+c(t)u= 0; and by assumption (ii)these functions are linearly independent. Hence
This is the end of the preview.
access the rest of the document.