Question

8. Consider the nonhomogeneous linear system of differential equations 1 1 1 -1 u = -1 11 1 1 u-et 1 1 2 3 First of all, find a fundamental matrix and the inverse matrix of the fundamental matrix of the corresponding homogeneous linear system. Then given a particular solution 71 uy(t) = et 1 2 find the general solution of the nonhomogeneous linear system of differential equations. Hint: det(A - \I) = -(1 – 2)?(1+1)

Answer 1

Sol A : Bt we Aind eigenvalue 1A-0 Hint detCA-Az) -CA-2Ca+ aet CA-Ar)-O -CA-23CA+10 eigen vectorsy B1meed fo solve CA- O Now we fnd -1 2 RRI/2 I 1 ROR+R,, R3 Ra-Ri 1 6 39 12 1/2 3t2 3 Ra Ra -2 1

Ri R+R Xl+33-0 = 1-3 Put x31 then -a -) V 2-2 we'll need to solue CA)-o CF -1 - 1 1 R2-)R2-RI, R3 Ra+R -1 R, RI/ -1 tY-NDO %-9+z (-y+z +lut) ge $gen-sol: ucth-ct)+upct +e