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Solving First-Order Differential Equations Using Laplace Transforms

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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:


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



  • 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?


  • 1. Eves, H., An Introduction to the History of Mathematics, Holt, Reinhart and Winston, New York, 1976.
  • 2. Owen, D.B., Editor, On the History of Statistics and Probability, Marcel Dekker, New York, 1976.
  • 3. Lipschutz, S., Set Theory, McGraw Hill, New York, 1964.
  • 4. Speigel, M.R., Probability and Statistics, McGraw Hill, New York, 1975.
  • 5. Speigel. M.R., Statistics, McGraw Hill, New York. 1961.
  • 6. Report No. NUREG-0492, Fault Tree Handbook, U.S. Nuclear Regulatory Commission, Washington, DC, 1981.
  • 7. Mann, N.R., Schefer, R.E., Singpurwalla, N.D., Methods for Statistical Analysis of Reliability and Life Data, John Wiley, New York. 1974.
  • 8. Lipschutz, S., Probability, McGraw Hill, New York, 1965.
  • 9. Spiegel, M.R., Laplace Transforms, McGraw Hill, New York, 1965.
  • 10. Oberhettinger, F., Baddii, L., Tables of Laplace Transforms, Springer-Verlag, New York, 1973.
  • 11. Patel, J.K., Kapadia, C.H., Owen, D.H., Handbook of Statistical Distributions, Marcel Dekker. New York, 1976.
  • 12. Davis, D.J., Analysis of some failure data, Journal of the American Statistical Association 1952, pp. 113-150.
  • 13. Weibull, W., A statistical distribution function of wide applicability, Journal of Applied. Mechanics, V'Ы. 18. 1951. pp. 293-297.
  • 14. Dhillon. B.S., Life distributions, IEEE Transactions on Reliability, Vol. 30, 1981, pp. 457^160.
  • 15. Baker, R.D., Non-parametric estimation of the renewal function, Computers Operations Research, Vol. 20, No. 2, 1993, pp. 167-178.
  • 16. Cabana, A., Cabana, E.M., Goodness-of-fit to the exponential distribution, focused on Weibull alternatives, Communications in Statistics-Simulation and Computation, Vol. 34. 2005, pp. 711-723.
  • 17. Grane, A., Fortiana, J., A directional test of exponentiality based on maximum correlations, Metrika, Vol. 73, 2011, pp. 255-274.
  • 18. Henze, N.. Meintnis, S.G., Recent and classical tests for exponentiality: a partial review with comparisons, Metrika, Vol. 61,2005, pp. 29-45.
  • 19. Jammalamadaka, S.R., Taufer, E., Testing exponentiality by comparing the empirical distribution function of the normalized spacings with that of the original data, Journal of Nonparametric Statistics, Vol. 15, No. 6, 2003. pp. 719-729.
  • 20. Hollander, M., Laird, G., Song, K.S., Non-parametric interference for the proportionality function in the random censorship model. Journal of Nonparametric Statistics, Vol. 15, No. 2,2003, pp. 151-169.
  • 21. Jammalamadaka, S.R., Taufer, E., Use of mean residual life in testing departures from exponentiality, Journal of Nonparametric Statistics, Vol. 18, No. 3,2006, pp. 277-292.
  • 22. Kunitz, H., Pamme, H., The mixed gamma ageing model in life data analysis, Statistical Papers, Vol. 34, 1993, pp. 303-318.
  • 23. Kunitz, H., A new class of bathtub-shaped hazard rates and its application in comparison of two test-statistics, IEEE Transactions on Reliability, Vol. 38, No. 3,1989, pp. 351-354.
  • 24. Meintanis, S.G., A class of tests for exponentiality based on a continuum of moment conditions. Kybernetika, Vol. 45, No. 6, 2009, pp. 946-959.
  • 25. Morris, K., Szynal, D., Goodness-of-fit tests based on characterizations involving moments of order statistics, International Journal of Pure and Applied Mathematics, Vol. 38, No. 1, 2007. pp. 83-121.
  • 26. Na, M.H., Spline hazard rate estimation using censored data. Journal of KSIAM, Vol. 3, No. 2, 1999, pp. 99-106.
  • 27. Morris, K., Szynal, D., Some U-statistics in goodness-of-fit tests derived from characterizations via record values, International Journal of Pure and Applied Mathematics, Vol. 46. No. 4, 2008. pp. 339-414.
  • 28. Nam, K.H., Park, D.H., Failure rate for Dhillon model, Proceedings of the Spring Conference of the Korean Statistical Society, 1997, pp. 114-118.
  • 29. Nimoto, N., Zitikis, R., The Atkinson index, the Moran statistic, and testing exponentiality. Journal of the Japan Statistical Society, Vol. 38, No. 2, 2008, pp. 187-205.
  • 30. Nam, K.H., Chang, S.J., Approximation of the renewal function for Hjorth model and Dhillon model, Journal of the Korean Society for Quality Management, Vol. 34, No. 1, 2006. pp. 34-39.
  • 31. Noughabi, H. A., Arghami, N.R., Testing exponentiality based on characterizations of the exponential distribution. Journal of Statistical Computation and Simulation, Vol. 1, First,
  • 2011,pp. 1-11.
  • 32. Szynal, D., Goodness-of-fit derived from characterizations of continuous distributions, Stability in Probability, Banach Center Publications, Vol. 90, Institute of Mathematics, Polish Academy of Sciences, Warszawa, Poland. 2010, pp. 203-223.
  • 33. Szynal, D., Wolynski, W„ Goodness-of-fit tests for exponentiality and Rayleigh distribution, International Journal of Pure and Applied Mathematics, Vol. 78, No. 5, 2012, pp. 751-772.
  • 34. Nam, K.H., Park, D.H., A study on trend changes for certain parametric families, Journal of the Korean Society for Quality Management, Vol. 23, No. 3, 1995, pp. 93-101.
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