Dsolve command

Maple s dsolve command can be used to obtain analytical and series solutions for differential equations. Differential equations are discussed in more detail in chapters 2 and 3. In this section, some Maple conunands are introduced to solve relatively simple differential equations. 

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. 

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. 

We observe that solutions obtained for T2(t) and T3(t) using the dsolve command are long and messy compared to the solution obtained using the exponential matrix approach (Example 2.3). When more than three differential equations are to be solved it is recommended that the exponential matrix method be used. As an exercise, readers can verily that the solution obtained using the dsolve command is equivalent to the solution obtained in example 2.3. 

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. 

Series solutions for nonlinear ODEs can be obtained using Maple s dsolve command. The syntax is ... 

Similarly higher order ODEs can be solved using Maple s dsolve command as shown in the next example. 

In this chapter, nonlinear IVPs were solved numerically. In section 2.2.2 a noniinear IVP was solved analytically using Mapie s dsolve command. This approach is limited to very few nonlinear ODEs. In section 2.2.3, series solutions were obtained using Maple s dsolve command. This approach is valid for all... 

Thus, the process yields a series solution in t for C. This solution can be compared to the series solution given by Maple s dsolve command ... 

Readers can verify this series solution with the series solution obtained by using Maple s dsolve command. 

In section 3.1.6, linear Boundary value problems were solved using Maple s dsolve command. The solution obtained may not be in the simplified form. Maple gives the Bessel function and other special function solutions for linear... 

Simultaneous reactions, mass/momentum transfer, heat/mass/momentum transfer are often represented by coupled boundary value problems. Coupled boundary value problems can be conveniently solved numerically using Maple s dsolve command. The syntax is ... 

Obtain the series solutions for this problem using Maple s dsolve command. (Since there is a removable singularity at x = 0, use c(l)=l and D(c)(l) = cl to obtain series solutions and obtain the constant cl using the boundary condition at x = 0). 

Next, the dependent variable in the Laplace domain is solved using the dsolve command since (see chapter 3.1.6) ... 

In this chapter, analytical solutions were obtained for parabolic and elliptic partial differential equations in semi-infinite domains. In section 4.2, the given linear parabolic partial differential equations were converted to an ordinary differential equation boundary value problem in the Laplace domain. The dependent variable was then solved in the Laplace domain using Maple s dsolve command. The solution obtained in the Laplace domain was then converted to the time domain using Maple s inverse Laplace transform technique. Maple is not capable of inverting complicated functions. Two such examples were illustrated in section 4.3. As shown in section 4.3, even when Maple fails, one can arrive at the transient solution by simplifying the integrals using standard Laplace transform formulae. 

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