Question

15. Consider the differential equation (*) х%3D Ах + vex where v is an eigenvector of A with eigenvalue X. Suppose moreover, that A has linearly independent eigenvectors v', v2,... , v", with distinct eigenvalues Ар А, гespectively. (a) Show that (*) has no solution /(t) of the form V(t)= ae*. Hint: Write a= а у+.. +a, v. (b) Show that (*) has a solution /(t) of the form n 0) -ае^ + bieM, 367

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Answer 1

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Page No. Date /20 AX=X is solectien of AX Cannt be a selutisn el i-aKtvet assluti on Ct ae +bte KARVet 73 Show ylE2aivet tavest t(b,'etthve anvettirts bnv'ent an n Vet + +tlby'e t2 v'e ha nt dhaly AVIE- A p?.t an Vent)t(b r'et ynent , E Au(t) anvnt tbvet ha?ayent A UCE) + b.v'et he b =74)0 YAUCts + Vert 30 UTESeluch on to t