Solving First-Order Differential Equations Using Laplace Transforms

8 Solving First-Order Differential Equations Using Laplace Transforms 

In safety studies, linear first-order differential equations are sometime solved. Laplace transforms are a useful tool for solving a set of linear first-order differential equations. The application of Laplace transforms to solve a set of linear first-order differential equations describing a patient safety problem in a health care facility is demonstrated through the following example. 

Assume that patient safety-related incidents occurring at a health care facility are described by the following two linear first-order differential equations  

Pi t) = the probability that there is no patient safety-related incident at the health care facility at time t. 

Laplace transforms are used to find solutions to linear first-order differential equations, particularly when a set of linear firsf order differential 

Assume that a transportation system can be in either of the three states operating normally, failed safely, or failed unsafely. The following three differential equations describe each of these transportation system states  

Pj(t) = probability that the transportation system is in state j at time f, for y = 0 (operating normally), y = 1 (failed unsafely), and j = 2 (failed safely) 

Using Table 2.1, the specified initial conditions, and Equations (2.55)-(2.57), we get 

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Equations (4-21) are linear first-order differential equations. We considered in detail the solution of such sets of rate equations in Section 3-2, so it is unnecessary to carry out the solutions here. In relaxation kinetics these equations are always solved by means of the secular equation, but the Laplace transformation can also be used. Let us write Eqs. (4-21) as... 

Equations D-6 and D-7 are first order differential equations with constant coefficients. One of the easiest ways to solve such equations is to use a Laplace Transform to convert from the time domain (t) to the frequency domain (s). Taking the Laplace Transforms ... 

Several other methods are widely used to derive complex rate equations. Sets of first order differential equations with constant coefficients can be solved by a variety of methods listed in standard textbooks. The use of second order differential equations for this purpose, as well as of the method of Laplace transforms, is illustrated in section 5.3. 

Many drugs undergo complex in vitro drug degradations and biotransformations in the body (i.e., pharmacokinetics). The approaches to solve the rate equations described so far (i.e., analytical method) cannot handle complex rate processes without some difficulty. The Laplace transform method is a simple method for solving ordinary linear differential equations. Although the Laplace transform method has been used for more complex applications in physics, engineering, and other research areas, here it will be applied to ordinary differential equations of first-order rate processes. 

Linear first order hyperbolic partial differential equations are solved using Laplace transform techniques in this section. Hyperbolic partial differential equations are first order in the time variable and first order in the spatial variable. The method involves applying Laplace transform in the time variable to convert the partial differential equation to an ordinary differential equation in the Laplace domain. This becomes an initial value problem (IVP) in the spatial direction with s, the Laplace variable, as a parameter. The boundary conditions in x are converted to the Laplace domain and the differential equation in the Laplace domain is solved by using the techniques illustrated in chapter 2.1 for solving linear initial value problems. Once an analytical solution is obtained in the Laplace domain, the solution is inverted to the time domain to obtain the final analytical solution (in time and spatial coordinates). This is best illustrated with the following example. 

In this section we will give two characteristic examples of solving linear differential equations using the Laplace transforms. The first example is the solution of a second-order differential equation, while in the second example we find the solution to a system of two differential equations. 

First order series/parallel chemical reactions and process control models are usually represented by a linear system of coupled ordinary differential equations (ODEs). Single first order equations can be integrated by classical methods (Rice and Do, 1995). However, solving more than two coupled ODEs by hand is difficult and often involves tedious algebra. In this chapter, we describe how one can arrive at the analytical solution for linear first order ODEs using Maple, the matrix exponential, and Laplace transformations. 

Parabolic partial differential equations are solved using the Laplace transform technique in this section. Diffusion like partial differential equations are first order... 

In the present calculation we solve the temperature equation (9.228) in terms of the flux ip(Zyt) and use this result in (9.229) and thence in (9.226). The temperature equation may be solved by the method of Laplace transforms as in the preceding analysis or by the general methods for treating first-order partial differential equations, It may be shown that the application of the latter method leads to the results... 

Let us consider an example of using the operator method for solving a differential equation describing a concentration change of an intermediate during a first-order consecutive reaction (Fig. 2.8). The original function here is the intermediate current concentration CB(t). A use of the Mathcad operator Laplace leads to the Laplace transform of this function in the form Laplace (CbIL), t,s). Pay attention to the result of the transform of the derivative Cb (t). Besides the... 

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