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Math 202-Di§erential Equations, Midterm 3 1

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Math 202-Di§erential Equations, Midterm 3

1. [2nd order and systems](14+21 points)Consider the second order ODE () y 00 + 2ay0 + by = 0; where a and b are real constants. (a) Show that () is equivalent to the system x 0 1 = x2; x 0 2 = bx1 2ax2: (b) Solve the system of di§erential equations for a = 1, b =

2. [Eigenvalues](15 points) Three solutions of the equation x 0= Ax are 0 @ e t + e 2t e 2t 0 1 A , 0 @ e t + e 3t e 3t e 3t 1 A and 0 @ e t e 3t e 3t e 3t 1 A . Find the eigenvalues and eigenvectors o

3. [Systems](8+20 points) Let A = 1 1 1 : (a) Determine all values of for which the matrix A has distinct real eigenvalues. (b) Suppose that is among the values found in part a). Solve the system x 0 = Ax; where A = 1 1 1 .

4. [Laplace transform](22 points)Use Laplace transforms to solve the given initial value problem. y 00 4y 0 + 5y = t; y(0) = 0; y0 (0) = 1: A Table of Laplace Transforms f(t) F(s) = L[f](s) e at 1 sa f (n) (t) s nF(s) s n1f(0) s n2f 0 (0) sf(n2)(0) f (n1)(0) sin(at) a s 2+a2 cos(at) s s 2+a2 t