Question

hi! I need help with this college level differential equations question. Please show all work and thank you.

3. We will next use matrix exponentials to find a fundamental matrix for the given system of DEs, (t) = P3(t) , P= (a) Letỉ(t) Ty(t), where Ț s a 2 2 constant matrix. Then, the system becomes = Dy, where D-T-1 PT. Find matrices T and D, such that D is a diagonal matrix. (Note that D and T are complex matrices.) (b) The new system ที่ Dy is diagonal. Find the fundamental matrix solution Y(t) eDt. You can calculate the matrix exponential et directly since the matrix is diag- onal. Equation (42) from Section 7.7 in the textbook may be helpful. c) Since D is complex, your solution Y (t) has complex exponentials inside. Use Eulars formula to rewrite complex exponentials as real exponentials and sin/cos functions. The imaginary unit i should still appear in Y(t), but not in exponentials anymore.) (d) Use your matrix T to find a fundamental matrix solution of the original system, x(t) ePt-TeDtT-1. (Note: the matrix solution X(t) should be real even if D and T are complex.) This process also tells you how to calculate matrix exponentials when the matrix is not diagonal.)

Answer 1

Sl. a) 2.

et e r LO 1+1 -1 141.101

t41 2 41 It に 4ゴ :, (..((@d*n)'stu(G"-/(.,$)(3)