Question

1) Solve the following second order linear ODEs by i. Finding the roots of differential operator on the Left Hand Side; ii. Finding the roots of the annihilator of the Right Hand Side of the equation; If the Right Hand Side is zero, there is nothing to do here! iii. Writing down a basis of the solution of the homogeneous equation; iv. Reducing the task of finding the undetermined coefficient to solving a system of linear equations in augmented matrix form by finding the column vector v for the Right Hand Side in terms of the basis and the matrix of differentiation A in terms of the basis. مترجم HINT: An E)-(0-2) Rootse" =1, m+1 times; "{cos (B1), cos(be)} = ((D-(a+Bi))(D-(a – B:))"; Rootsedes {cos(pt), cos(pt)} = a + ßi, m+1 times; HINT: Ann m! m! e)((D-(7+8i))(D-(7 – 8i))) = 71%e3"; f) ((D-(7+8i))(D-(7-8i))) = 7te* cos(5e).

Can you show parts i through iv for parts e and f. please and thank you.

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Can you please show the second-order linear ordinary differential equations (ODE) for parts e and f by answering parts i through iv for e and i through iv for f. Thank you.

Answer 1

part (e) = ((0-16+31 1)(0-47-81)))76e2t of left hand side - Gluens 7th ezt compare with th 2 part It) at Casst (1) Root of DES for port (e) D= 7+81, 7+81 and 7-8, 7-8 Hence Poots are- 78, 7+8i for part (t)! O = 7+31, 7+Bi and 7-8i, 7-81 Hence roots are D=78i, 781 (ii) Root of DES for part (es! =2 2 of Right hand side- ett mi Roots are 2; 2,2

for part ti- tezt m Casst compose with t ext eaten Bt ml Here =2, d=0, B=5, and m=2. then Roots are- Roots elsea 75, IS, IS All) Basis of solution of ODES part (e) 7t Y = e (638t + i Singt) #yetett (les st+isinet) and part (t):- 가 YI (los 8t + i sinet), Y2=te — е 7tos&ttisin st) (iv) portle) Interms of argumented matrix. 7+2 e 2x 7tos st