Applied Engineering Analysis
                                        - slides for class teaching*
                                                  Chapter 7
                    Application of First-order Differential Equations 
                                       in Engineering Analysis
                              * Based on the book of  “Applied Engineering  
                                 Analysis”, by Tai-Ran Hsu, published by
                                 John Wiley & Sons, 2018
           (Chapter 7 First order DEs)
           © Tai-Ran Hsu
     Chapter Learning Objectives
     ●Learn to solve typical first order ordinary differential equations of both 
     homogeneous and non‐homogeneous types with or without specified conditions.
     ●Learn the definitions of essential physical quantities in fluid mechanics analyses.
     ●Learn the Bernoulli’s equation relating the driving pressure and the velocities of 
     fluids in motion.
     ●Learn to use the Bernoulli’s equation to derive differential equations describing 
     the flow of non‐compressible fluids in large tanks and funnels of given geometry.
     ●Learn to find time required to drain liquids from containers of given geometry 
     and dimensions.
     ●Learn the Fourier law of heat conduction in solids and Newton’s cooling law for 
     convective heat transfer in fluids.
     ●Learn how to derive differential equations to predict required times to heat or cool 
     small solids by surrounding fluids.
     ●Learn to derive differential equations describing the motion of rigid bodies under the 
     influence of gravitation.
             7.1 Introduction on Differential Equations
                 Types of Differential equations:
                 We have learned in Chapter 2 that differential equations are the equations that involve 
                 “derivatives.” 
                 They are used extensively in mathematical modeling of engineering and physical problems.
                 There are generally two types of differential equations used in engineering analysis. These are:
                 1.   Ordinary differential equations (ODE): Equations with functions that involve only one 
                      variable and with different orders of “ordinary” derivatives , and
                 2.   Partial differential equations (PDE): Equations with functions that involve more than one 
                      variable and with different orders of  “partial” derivatives.
                 How differential equations are derived?
                 They are derived from the three fundamental laws of physics of which most engineering 
                 analyses involve. These laws are: 
                 (1) The law of conservation of mass,
                 (2) The law of conservation of energy, and 
                 (3) The law of conservation of momentum. 
        7.1 Introduction on Differential Equations – Cont’d
        Differential equations for mechanical engineering:
        For mechanical engineering analyses, frequently used laws of physics include the 
        following:
        ●The Newton’s laws for statics, dynamics and kinematics of solids.
        ●The Fourier’s law for heat conduction in solids.
        ●The Newton cooling law for convective heat transfer in fluids.
        ●The Bernoulli’s principle for fluids in motion.
        ●Fick’s law for  diffusion of substances with different densities
        ●Hooke’s law for deformable solids