# real and complex eigenvalues and eigenvectors

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• using Euler’s Method to solve an ODE. • computing and working with real and complex eigenvalues and eigenvectors. • solving a system of 1st order ODEs as an eigensystem.
Consider the initial value problem dy x cos (7 r Y ) = 1 with y(-2) = 0. dx 2Use Euler’s method with Ax = 1 to estimate y when x = 4.
Solve the initial value problem y’ (t) = A y (t) with [ —2-5 A = 0 , Y(0) = 1 1and y (t) =
Consider the homogeneous 2nd order ODE X” + 2X’ + 5X = 0 with initial conditions X(0) = 1 and X’ (0) = 2. Assuming X(t) = e)t, compute A then find X(t) as a linear combination of two independent solutions.
Show that the 2nd order ODE in question 3 can be transformed into the system of 1st order ODEs in question 2 with the substitutionsyi (t) = X’ (t) and y 2(t) = X (t) .