Solving First-Order Differential Equations Using Laplace Transforms

Generally, Laplace transforms are used for finding solutions to linear first-order differential equations, particularly when a set or linear first-order differential equations is involved. The example presented below demonstrates the finding of solutions to a set of linear first-order differential equations, describing a transportation system in regard to reliability, using Laplace transforms.

Example 2.10

Assume that a transportation system can be in any of the three states: operating normally, failed due to a hardware failure, or failed due to a human error. The following three first-order linear differential equations describe each of these transportation system states:

where

P, (?) is the probability that the transportation system is in state i at time t, for i = 0 (operating normally), i = 1 (failed due to a hardware failure), and i = 2 (failed due to a human error).

Ял is the transportation system constant hardware failure rate.

A;„, is the transportation system constant human error rate.

By using Table 2.1, differential Equations (2.52)-(2.54), and the given initial conditions, we obtain:

By solving Equations (2.55)-(2.57), we get

By taking the inverse Laplace transforms of Equations (2.58)-(2.60), we obtain

Thus, Equations (2.61)-(2.63) are the solutions to differential Equations

(2.52)—(2.54).

Problems

  • 1. Quality control department of a transportation system manufacturing company inspected six identical transportation systems and discovered 5, 2, 8, 4, 7, and 1 defects in each respective transportation system. Calculate the average number of defects (i.e., arithmetic mean) per transportation system.
  • 2. Calculate the mean deviation of the Question 1 dataset.
  • 3. What is idempotent law?
  • 4. Define probability.
  • 5. What are the basic properties of probability?
  • 6. Assume that an engineering system has two critical subsystems Xx and X2. The failure of either subsystem can, directly or indirectly, result in an accident. The failure probability of subsystems Xx and X, is 0.06 and 0.11, respectively.

Calculate the probability of occurrence of the engineering system accident if both of these subsystems fail independently.

  • 7. Define the following terms concerning a continuous random variable:
    • • Cumulative distribution function
    • • Probability density function
  • 8. Define the following two items:
    • • Expected value of a continuous random variable
    • • Expected value of a discrete random variable
  • 9. Define the following two items:
    • • Laplace transform
    • • Laplace transform: final-value theorem
  • 10. Write down the probability density function for the Weibull distribution. What are the special case probability distributions of the Weibull distribution?

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