Modeling Tools for Environmental Engineers and Scientists

SolutionThe solution can be found by first categorizing the ODE and then followingstandard appropriate mathematical calculi. Alternatively, some computersoftware packages can find the analytical solution directly. The two approachesare illustrated in this example.The given equation can be seen as a special case of a first-order nonhomogeneouslinear equation identified earlier, with y = C, x = t, P(x) = 1.23,and Q(x) = 2.0. It can, therefore, be solved by applying Equation (3.18):Integrating factor = ≡ e ∫P(t)dt ≡ e ∫1.23dt = e 1.23tHence, the solution to ODE is as follows:2 1. 23 {e1.23t } b∫{e 1.23t 2}dt bC = = e 1.23 e1.23tt2= 1. 23 1 1. 232 be –1.23t From the initial condition given, C = 0 at t = 0; therefore, b = – 1.223 Thus, the final solution for C as a function of t is as follows:C 1.223 {1 – e–1.23t } 1.63{1 – e –1.23t }See Worked Example 3.4 for a plot of the above result.As an alternate approach, an equation solver-based software package,Mathematica ® , is used to find the solution directly as shown in Figure 3.9.Here, the built-in procedure, DSolve, is called in line In[1] with the followingarguments: the ODE to be solved, the initial condition, the dependentvariable, and the independent variable. The solution is returned in line Out[2],with the integration constant automatically evaluated, by Mathematica ® .Finally, the Plot function is called to plot the solution showing the variationof C with t.It can be seen that the solution returned by Mathematica ® in line Out[2],after some manipulation, is identical to the solution found earlier by followingthe standard mathematical calculi. This example illustrates just one of thebenefits of such packages in easing the mathematical tasks involved in modeling,enabling modelers to focus more on formulating and posing the problemat hand in standard mathematical forms rather than on the operandi ofsolving the formulation.© 2002 by CRC Press LLC