Mathematics 1229A/B Midterm: review for mid 2

Problem 7 (Solving system of ODE). Consider a certain system of two first order linear differential equations in two unknowns, x' = Ax, where A is a matrix of real numbers. Suppose one of the eigenvalues of the coefficient matrix A is r = 3 + i, which has a corresponding eigenvector ( ). What is the system's real valued general solution? Problem 8 (Stability of solution). For what value(s) of beta will the linear system below have a (neutrally) stable center at (0, 0)? X?=

Problem 4 (Type and Stability). For each part below, consider a certain system of two first order linear linear differential equations in two unknowns, x? = Ax, where A is a 2 x 2 matrix of real numbers. Based solely on the information given in each part, determine the type and stability of the systems critical point at (0, 0).

3. A linear system containing three equations in the seven variables r1,2, 3, 24,z5, 6 and a7 has an augmented matrix of the following form 0 0.* 0 0 0 0 0 * In the above matrix, the symbol represents a nonzero number entry and the symbol represents a number entry that may be positive negative or 0. Is the system consistent or inconsistent? If it is consis- tent, indicate whether the system has a unique solution or infinitely many solutions, explaining your reasoning. If it has infinitely many solutions, indicate which are the free variable and which are the basic variables.