Write the following regarding solutions of a system → x′= A → x: 1. The deﬁnition...

Write the following regarding solutions of a system
→ x′= A → x: 1. The deﬁnition of a Fundamental Matrix Φ(t), and apply it in one example of your choosing (for a constant matrix A that you pick). 2. On your example in part 1. calculate the function y(t) = detΦ(t) (the determinant of Φ(t) and check that y(t) solves the ﬁrst order ODE y′(t) = (trA)y(t) (the trace (trA) is the sum of the entries on the main diagonal of A). 3. Choose a constant 2×2 matrix A different from the one used before. By solving the ODE y′(t) = (trA)y(t) conclude that y(t) is either identically zero, or is never zero.

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Colby Messinger answered 1 year ago

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