Solved: In 13 and 14, you are given a nonhomogeneous second-order linear differential

Chapter 7, Problem 13

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In 13 and 14, you are given a nonhomogeneous second-order linear differential equation and two linearly independent solutions, y1 and y2, to the corresponding homogeneous differential equation. Use this information to complete the following steps: a. Find the equivalent nonhomogeneous system of first-order linear differential equations for x1 = y and x2 = y. b. Show that x(1) = ( y1, y 1) T and x(2) = ( y2, y 2) T are solutions to the homogeneous system of differential equations corresponding to the system found in a. (As a consequence, = x(1) | x(2) is a fundamental matrix for the same homogeneous system.) c. Find the variation of parameters equations that have to be satisfied for y = y1(t)u1(t) + y2(t)u2(t) to be a particular solution of the given nonhomogeneous second-order differential equation. d. Find the variation of parameters equations that have to be satisfied for x = (t)u(t) to be a particular solution of the nonhomogeneous system of first-order linear differential equations found in a. e. Use the definition of x(1) and x(2) in b to show that the systems of equations found in c and the equations found in d are equivalent.y__5y_+6y = 2et , y1 = e2t , y2 = e3t ( 1, Section 3.6) 1