1. Consider the following ODE: u'' + 5u' - 24u = 0.
a. Convert the second order ODE into a system of 2 first order ODEs in to the form of iT=Au.
b. Find the eigenvalues and eigenvectors of matrix A.
2. Find the characteristic equation and the corresponding roots for the following ODEs and identify its case
a. y'' + 8y' + 16y = 0
b. 3y'' - 18y' + 75y = 0
c. 6y'' + 17y' + 10y = 0.
3. Solve the initial value problems (i. e. find particular solutions)
a. Problem 2a. with y(0)=2 and y'(0)= -5.
b. Problem 2b. with y(0)= 1. 5 and y'(0)= 14.5.
c. Problem 2c. with y(0)= -7 and>''(0)=l.
4. a. Write the general solution for problem 1 in terms of eigenvalues and eigevectors obtained in the problem.
b. Find The particular solution for problem I if u(0)=3 and u'(0)=2.