Exercise 3.3.2: Verify that the system x® 0 1 3 3 1 x® has the two solutions 1 1 e 4t and 1 −1 e −2t a) . b) Write down the general solution. Write down the general solution in the form x1 ?, x2 ? (i. e. write down a formula for each element of the solution).

The general solution of the system is

Step-by-step explanation:

To verify that the system has the solutions given,

First, we determine the Eigen Values β of the matrix of the form Using |A-βI|=0, where I is the Identity Matrix, we have:

Subtracting matrices

Taking the determinant

We determine the eigen vector using

When = -2,

which implies that and thus  

When = 4,

which implies that and thus  

A general solution of the system is given as

where the c's are the eigen vectors.

Thus the general solution is

problem:

f(x) = - 2/3x + 5; f(5/2)

substitute 5/2 in for x:

f(5/2) = -2/3(5/2) + 5

simplify the right side:

f(5/2) = -5/3 + 5

f(5/2) = 10/3

f(5/2) = 10/3

i hope this !

i believe the correct answer is b.