Question

Show that if the linear homogeneous system Y'=AY' has a solution of the form Y'(t)=eλᵗv', then Y'(t)=eλᵗ(kv') is also a solution for any constant k

Answer 1

It is shown that Y'(t) = eλt(kv') is a solution to the given linear homogeneous system by applying the eigenvalue relationship involving the matrix A and the principle of superposition.

Step-by-step explanation:

The question revolves around showing that if Y'(t) = eλtv' is a solution to the system Y' = AY', then Y'(t) = eλt(kv') is also a solution for any constant k. This can be understood by acknowledging the principle of superposition, which applies to linear homogeneous systems. If Y'(t) = eλtv' is a solution, this means that AY'(t) = A(eλtv') = eλtAv' = λeλtv' (since Av' = λv' for some eigenvalue λ). Multiplying both sides of this equation by k, we obtain kλeλtv' = A(keλtv'), showing that Y'(t) = eλt(kv') is also a solution.