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PROBLEMS: Section 5

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PROBLEMS: Section 5.6 For Problems 1-12, solve the given initial-value problem.sketch the graph of both the forcing function and the solution. 1. y'=1 - u(t-1) y(0)=0 2. y'=1 - 2u(t-1) +ult - 2) y(0=0 3. y' + y = u(t-1) 4. y" + y =t-u(t-1) y(0)=0 y'(0) = 0 5.y" = 1-u(t-1) y(0=0 y'(0) = 0 6. y" + y = u(t-3) yO=0 y'(0) = 1 7. y" + y = u(t-T) - ult-27) y(0)=0 y'(0)=0 y" + 4y = -- ) 8. y(0=0 y'(0)=0 9. y" + 4y = u(t-21)sint y(0-1 y'(o)=0 10. y" + 4y = sin t-u?t-21) sin(t-21) y(0=0 y'(0) = 0 11. y" + y +3y= ult-2) y(O)=0 y'(0) = 1 12. y" + 2y' +5y = 10-10u(t-) y(0)=0 Aa 510 . PROBLEMS: Section 5.7 For Problems 1–4, "plot" the given function and find the Laplace transform of the given function. 1. 8(t - 1) + 28(1 - 2) + 38(1 - 3) 2. 8(t) - 28(t – T) + 8(t – 27) 3. 8(t) – 8(t – 1) + (t - 2) - 4. 8(t) + 8(t – T) + 8(t - 27) + ... For Problems 5-13, find the solution of the given initial-value problem. If you have access to a computer with a graphing package, sketch the graph of the solution. y' = 6t) 5. y(0) = 0 y' = 8(t) – 8(t – 1) 6. y(0) = 0 y" + y = $(t – 27) 7. y(0) = 0 y'(0) = 0 y" + y = -80t – T) + 8(t – 2TT) 8. y(O) = 0 y'(0) = 1 y" + y = 8(t – 27) 9. y(O) = 1 y'(0) = 0 y" + 4y' + 5y = 8(t) 10. y(0) = 0 y'(0) = 0 y" + 2y + 2y = 6t - T) 11. y(0) = 1 y'(0) = 0 y" + 2y' + 2y = 8(1 – T) 12. y(O) = 0 y'(0) = 0 y" + 4y' + 5y = 8(t 8(t – T) + 8(1 – 27) 13. y(0) = 0 y'(0) = 2 14. Spaceship Driven by a Series of Thrusts A spaceship with mass m is moving through frictionless space, driven only by a rocket that fires in short bursts of magnitude A at times a, 2a, 3a,.... If the initial position and velocity of the spaceship are y(O) = y(O) = 0, find the position of the spaceship at any time. Sketch the graph of the solution. 15. Differentiation in tEquals Multiplication by s We start with the integro-differential equation y+3y we dt = ut - 1 +C