2. (16 points) Consider the non- homogeneous first-order linear differential system V = yı + y2 4 = y1 + y2 Y = ya + y3 - 1 where the y,'s are all functions of the variable z. Which vector functions Ý are a solution to the system, if any.

Match the third order linear equations with their fundamental solution sets. L1. y" – 7y" + 10y' = 0 2. y" +y = 0 3. у" - 5у" +у —5у %3D0 _4. ty" – y' = 0 _5. y" – y" – y' +y=0 _6. y" +3y" +3y +y=0 | - A. {e-', te-', t²e-1} B. {1, cos(t), sin(t)} C. {e', te' , e-'} D. {1, et, e² } E. {e", cos(t), sin(t)} F. {1, t, 1³ }

Convert the Second order ODE  x''+2x'+9x+5=0 to a first order system of two equations. Use y for x'. x'(t) = ? y'(t)= ?

A. y" + 2y' + y = 0 B. y" + 2y' + 2y = 0 C. y" + 4y = 0 D. y" – 3y' + 2y = 0 Table 1: A-D are homogeneous second order linear equations.

A. y" + 2y' + y = 0 B. y" + 2y' + 2y = 0 C. y" + 4y = 0 D. y" – 3y' + 2y = 0 Table 1: A-D are homogeneous second order linear equations.

Transform the given system of differential equations into an equivalent system of first-order differential equations. x' - 3x' - 5x+2y=0 y'' +6y' - 6x-3y = sin t Let x₁ = x, X₂ = X', Y₁ = y, and y₂ = y'. Complete the system below. X₁' X₂ X Y2 II 11 11 11

5. Given the system X" - ()* 10 2 (a) Find the general solution. (b) Find the solution that satisfies the initial condition (x(0), y(0)) = (0, –7). Report your solution as one vector.