4 Chapter in Review Answers to selected odd-numbered problems begin on page ANS-11.

In Problems 1 and 2, use the definition of the Laplace transform to find {f(t)}.

  2. f(t) =

In Problems 3–36, fill in the blanks or answer true/false.

  1. If f is not piecewise continuous on [0, ∞), then {f(t)} will not exist.
  2. The function f(t) = (et)10 is not of exponential order.
  3. F(s) = s2/(s2 + 4) is not the Laplace transform of a function that is piecewise continuous and of exponential order.
  4. If {f(t)} = F(s) and {g(t)} = G(s), then −1{F(s)G(s)} = f(t)g(t).
  5. {e−7t} =
  6. {te−7t} =
  7. {sin 2t} =
  8. {e−3t sin 2t} =
  9. {t sin 2t} =
  10. {sin 2t (t − π)} =
  11. =
  12. =
  13. =
  14. =
  15. =
  16. =
  17. =
  18. =
  19. =
  20. =
  21. =
  22. =
  23. =
  24. =
  25. =
  26. −1 =
  27. −1 =
  28. −1 =
  29. {e−5t} exists for s > .
  30. If {f(t)} = F(s), then {te8t f(t)} = .
  31. If {f(t)} = F(s) and k > 0, then {eat f(t− k) (t − k)}= .
  32. = whereas = .
  33. =
  34. =
  35. Given that use (6) of Section 4.2 to find
  36. Given that use (9) of Section 4.4 to find

In Problems 39–42, use the unit step function to write down an equation for each graph in terms of the function y = f(t) whose graph is given in FIGURE 4.R.1.

A curve labeled y equals f(t) is graphed on the t y plane. It starts at a point on the positive y axis, goes down and to the right, and reaches a low point in the first quadrant. Then, it goes up and to the right, reaches a high point, again goes down and to the right, and ends at the right of the first quadrant. A point on the curve just to the right of the low point corresponds to the t value of t subscript 0.

FIGURE 4.R.1 Graph for Problems 39–42

  1. Two pieces of a function are graphed on the t y plane. The first piece is a straight line that starts at the origin, goes to the right, and ends at the point (t subscript 0, 0). The second piece is a curve that starts at a point in the first quadrant that corresponds to the t value of t subscript 0, goes up and to the right, and reaches a high point. Then it goes down and to the right, and ends at the top right of the first quadrant.

    FIGURE 4.R.2 Graph for Problem 39

  2. Two pieces of a function are graphed on the t y plane. The first piece is a curve that starts at a point on the positive y axis, goes down and to the right, and reaches a low point. Then it goes slightly up and to the right, and ends at a point in the first quadrant that corresponds to the t value of t subscript 0. The second piece is a straight line that starts at the point (t subscript 0, 0), goes to the right, and ends at the right on the positive t axis.

    FIGURE 4.R.3 Graph for Problem 40

  3. Two pieces of a function are graphed on the t y plane. The first piece is a straight line that starts at the origin, goes to the right, and ends at the point (t subscript 0, 0). The second piece is a curve that starts at a point in the first quadrant that corresponds to the t value of t subscript 0, goes down and to the right, and reaches a low point. Then it goes up and to the right, reaches a high point, goes slightly down and to the right, and ends at the right of the first quadrant.

    FIGURE 4.R.4 Graph for Problem 41

  4. Three pieces of a function are graphed on the t y plane. The first piece is a curve that starts at a point on the positive y axis, goes down and to the right, and ends at a point in the first quadrant that corresponds to the t value of t subscript 0. The second piece is a straight line that starts at the point (t subscript 0, 0), goes to the right, and ends at the point (t subscript 1, 0). The third piece is a curve that starts at a point in the first quadrant that corresponds to the t value of t subscript 1, goes up and to the right, then goes down and to the right, and ends at the right of the first quadrant. The starting point of this piece is above the endpoint of the first piece.

    FIGURE 4.R.5 Graph for Problem 42

In Problems 43–46, express f in terms of unit step functions. Find {f(t)} and {et f(t)}.

  1. Three pieces of a function are graphed as straight lines on the t f(t) plane. The first piece starts at the origin, goes up and to the right, and ends at the point (1, 1). The second piece starts at the point (1, 1), goes to the right, and ends at the point (4, 1). The third piece starts at the point (4, 0), goes to the right, and ends at the right on the positive t axis.

    FIGURE 4.R.6 Graph for Problem 43

  2. Three pieces of a function are graphed on the t f(t) plane. The first piece is a straight line that starts at the origin, goes to the right, and ends at the point (pi, 0). The second piece is a curve that follows a wave pattern. It starts at the point (pi, 0), goes down and to the right, and reaches a low point (3 pi over 2, negative 1). Then it goes up and to the right through the point (2 pi, 0), reaches a high point (5 pi over 2, 1), goes down and to the right, and ends at the point (3 pi, 0). The curve is labeled y equals sin t, pi less than or equal to t less than or equal to 3 pi. The third piece is a straight line that starts at the point (3 pi, 0), goes to the right, and ends at the right on the positive t axis.

    FIGURE 4.R.7 Graph for Problem 44

  3. Three pieces of a function are graphed as straight lines on the t f(t) plane. The first piece starts at the point (0, 2), goes to the right, and ends at the point (2, 2). The second piece starts at the point (2, 2), goes up and to the right, and ends at the point (3, 3). The third piece starts at the point (3, 3), goes up and to the right, and ends at the top right of the first quadrant. The point (2, 2) is marked with a dot. The point (3, 3) is marked with a dot and labeled (3, 3).

    FIGURE 4.R.8 Graph for Problem 45

  4. Three pieces of a function are graphed as straight lines on the t f(t) plane. The first piece starts at the point (0, 0), goes up and to the right, and ends at the point (1, 1). The second piece starts at the point (1, 1), goes down and to the right, and ends at the point (2, 0). The third piece starts at the point (2, 0), goes to the right, and ends at the right on the positive t axis.

    FIGURE 4.R.9 Graph for Problem 46

In Problems 47 and 48, sketch the graph of the given function. Find {f(t)}.

In Problems 49–58, use the Laplace transform to solve the given equation.

  1. y″ − 2y′ + y = et, y(0) = 0, y′(0) = 5
  2. y″ − 8y′ + 20y = tet, y(0) = 0, y′(0) = 0
  3. y″ + 6y′ + 5y = t − t (t − 2), y(0) = 1, y′(0) = 0
  4. y′ − 5y = f(t), where f(t) = , y(0) = 1
  5. , where the graph of is given in FIGURE 4.R.10
    Four pieces of a function are graphed as straight lines on the t f(t) plane. The first piece starts at the point (0, 0), goes to the right, and ends at the point (1, 0). The second piece starts at the point (1, 0), goes up and to the right, and ends at the point (2, 1). The third piece starts at the point (2, 1), goes down and to the right, and ends at the point (3, 0). The fourth piece starts at the point (3, 0), goes to the right, and ends at the right on the positive t axis.

    FIGURE 4.R.10 Graph for Problem 53

  6. where

In Problems 59 and 60, use the Laplace transform to solve each system.

  1. x′ + y = t
    4x + y′ = 0
    x(0) = 1, y(0) = 2
  2. x″ + y″ = e2t
    2x′ + y″ = −e2t
    x(0) = 0, y(0) = 0
    x′(0) = 0, y′(0) = 0
  3. The current i(t) in an RC-series circuit can be determined from the integral equation

    where E(t) is the impressed voltage. Determine i(t) when R = 10 Ω, C = 0.5 F, and E(t) = 2(t2 + t).

  4. A series circuit contains an inductor, a resistor, and a capacitor for which L = h, R = 10 Ω, and C = 0.01 F, respectively. The voltage

    is applied to the circuit. Determine the instantaneous charge q(t) on the capacitor for t > 0 if q(0) = 0 and q′(0) = 0.

  5. A uniform cantilever beam of length L is embedded at its left end (x = 0) and is free at its right end. Find the deflection y(x) if the load per unit length is given by

  6. When a uniform beam of length L is supported on an elastic foundation the differential equation for its deflection is given by

    ,

    where k is the modulus of the foundation and −ky is the restoring force of the foundation that acts in the direction opposite to that of the load w(x). For algebraic convenience, suppose the differential equation is written as

    ,

    where . Assume and Use the Laplace transform and the table of Laplace transforms in Appendix C to find the deflection of a beam that is supported on an elastic foundation when the beam is simply supported at both ends and the load is a constant uniformly distributed along its length. See FIGURE 4.R.11. [Hint: You will also need the factorization

    A beam of uniform length L is simply supported at both ends. The beam is supported on an elastic foundation. A vertical line along the support at the left end is labeled y. A horizontal line along the axis of the beam is labeled x. An uniformly distributed load w(x) equals w subscript 0 acts along the entire length of the beam.

    FIGURE 4.R.11 Beam in Problem 64

  7. Use the Laplace transform and the table of Laplace transforms in Appendix C to find the deflection of a beam that is supported on an elastic foundation as in Problem 64 when the beam is embedded at both ends and the load is a constant concentrated at .