(9) (a) If f(t) = x1e −kt cos(ωt) + x2e −kt sin(ωt), then f'(t) has the...

(9) (a) If f(t) = x₁e ^−kt cos(ωt) + x₂e ^−kt sin(ωt), then f'(t) has the same form with coefficients y₁, y₂.

(i) Find the matrix A such that y = Ax.

(ii) Find the characteristic polynomial of A.

(iii) What can you say about eigenvalues of A? (iv) Interpret your answer to

(iii) as a calculus statement. That is, explain how your answer to (iii) could have been predicted from a basic fact of calculus.

(b) If f(t) = x₁ sin(t) + x₂ cos(t) + x₃tsin(t) + x₄t cos(t), then f'(t) has the same form with coefficients y1, ..., y4. Same questions (a)-(d) as previously.

Solutions

Expert Solution

Colby Messinger answered 1 month ago

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