Solving linear ODEs using Maple

8 Solving Linear ODEs Using Maple s dsolve Command 

In the previous sections we solved linear ODEs using exponential matrix (section 2.1.2 - 2.1.4) and the Laplace transform technique (section 2.1.5). Alternatively, Maple s dsolve command can be used to solve linear ODEs. However, the analytical solution obtained from the dsolve command may not be in a simplified form. 

The reaction scheme described in example 2.1 is solved below using the dsolve command. 

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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. 

Maple s dsolve command was used to solve linear ODEs in section 2.1.6. In our opinion, exponential matrix method is the best method to arrive at an elegant analytical solution. The Laplace transform technique illustrated in section 2.1.5 could be used for integro-differential equations. Maple s dsolve command has to be used if the exponential matrix method fails. 

All the tanks are at an initial temperature of 20°C. Find the time it will take for the third tank to reach 99% of its steady state value. The values of the constants are W = 100 kg/min, M = 1000 kg, Cp = 2kJ/kg°C, and UA = 10kJ/min°C. Determine how this time varies with the parameters a and p. Equations (equation (2.17)) can be solved using Maple and the procedure described above for nonhomogeneous simultaneous linear ODEs follows. 

Higher order linear ODEs can also be solved using the dsolve command. It should be noted that Maple solves equations in symbolic form. Therefore, even if the constants are numerical, the output is in symbolic form. Sometimes, this output can be messy. It should be noted that when more than two equations are solved the dsolve command may not be able to give an elegant solution. For illustration, the heat transfer problem solved in example 2.3 is solved below using Maple s dsolve command. 

Using the boundary conditions (equations (5.54) and (5.55)) the boundary values uo and un+i can be eliminated. Hence, the method of lines technique reduces the nonlinear parabolic PDE (equation (5.48)) to a nonlinear system of N coupled first order ODEs (equation (5.52)). This nonlinear system of ODEs is integrated numerically in time using Maple s numerical ODE solver (Runge-Kutta, Gear, and Rosenbrock for stiff ODEs see chapter 2.2.5). The procedure for using Maple to solve nonlinear parabolic partial differential equations with linear boundary conditions can be summarized as follows ... 

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