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Problem 8 Let a(t) =/= b(t) be given. The factorization for a second order ODE is...

Problem 8 Let a(t) =/= b(t) be given. The factorization for a second order ODE is commutative if (D + a(t) I) (D + b(t) I) y = (D + b(t) I) (D + a(t) I) y.

• Find condition on a(t) and b(t) so that the factorization is commutative.

• Find the fundamental set of solutions for a second order ODE that has a commutative factorization.

• Use the above results to find the fundamental set of solutions of y'' + 4 y' + 4 y = 0.

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All the three parts are done separately

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