SUNY Buffalo State Ordinary Differential Equations Questions

Please answer the following questions and show the work.

” A textbook on Ordninary Differential equations” by Shair Ahmad and Antonio Ambrosetti may help.

1.(25 points) Solve the solutions of X’ = AX with
1 0 2
0 0 1
0 1 0
X.
=
TX
2. (25 points) Find all eigenvalues {^x} and the corresponding eigenfunctions
{z}} for the following eigenvalue problem
(
x” – 2.r’ + (1+1)x = 0
x(0) = x(T) = 0
3. (30 points) Use the Frobenius method to find all solutions at the regular
singular point t = 0,
tx” – 2+² x’ + (t? + – x = 0,
4.
t >0.