Solve the system x1 x2 x10 0 x2 x1 x20 1SolutionThe give

Solve the system x\'1 = x2 x1(0) = 0 x\'2 = x1 x2(0) = 1

Solution

The given system can be written as

Ax =B where A = 0 1

                            -1 0

Eigen values of A are

-c(-c)+1 =0

c =1, -1

Eigen vectors for 1:

-x1+x2 =0 or x1 = x2 Hence eigen vector =(1 1)

Eigen vectors for -1:

x1+x2 =0 or (1,-1)

Hence solution is

x = c1e^t(1 1) + c2e^-t(1 -1)

x1(0) = 0

gives c1+c2 =0

x2 (0) = 1 gives

c1-c2 =1

Solving c1 = 1/2 and c2 = -1/2

Solution is

x = 1/2{e^t(1 1) + c2e^-t(1 -1)}