Question

Use the method of variation of parameters to solve the initial value problem x' = Ax + f(t), x(a) = x, using the following values. 1-1831)[ 9

Answer 1

The givew mitial valve problem is AR+FO) K na=na. 14-5T 15-2) where A- T 174 1 22362r| *00=1] att -estett he we betaget -Det the homogeneous we first frud on fundamental matrix for som ergen value det 4-2 -3 15 - 2- --(4-2)(242) +5=0 >> - 8+e9-49 +2+5 = 0 => A-29-3=0 - 9-39+-3=0 A (A-3) +1 (9-3) = 0 (-)1) = 0 21-9S Ergen vector, Let * =3 Then and v= (*) Ar=30 [$-+][9] - » [9] - a-b=0 29=3 Hence ergew vector

As let v = (0) Av=-12) 50-6=0 39 6=5a. Head ergeu vector eBt TE (t)= | Lest set I det X(t) = 5 eteet = 4e2t . Fetest 70 4e2 -83E est , - & 1 5 6 3t 4 L L et et Their we know that the solu b given by x) = x(!) c + xcel 'ct) {cu dt where an [at] we are cont

NOW, **() Fodt 83 02€ 103 Heu ce xmp 737+ [* ][9]+[ -[*]:13 the bittal cruda is x0) -(3) Mence etc Hence 4+C =.35 e. +56 = 9 (1) - 12

40 = Hence, G-355 -32 = 13. Hewer c-() Frualy. 35.RE