Closedform Solutions of Linear Differential Equations
The Maple dsolve command allows determination of closedform solutions for linear differential equations.

Secondorder Equations


The dsolve command converts a homogeneous secondorder linear differential equation of the form
into one of the form
, and then uses solutions of the new equation to build solutions to the first. The method computes a solution to the second by looking at the partial fraction expansion of I(x).
For example, we have (from the classical text by Kamke):
>


>


 (1.1) 
>


 (1.2) 
>


 (1.3) 
>


 (1.4) 
>


 (1.5) 
>


 (1.6) 
>


 (1.7) 
>


 (1.8) 
In all of these examples, one can verify the solutions by using the odetest command:
>


 (1.9) 
>


 (1.10) 
>


 (1.11) 
>


 (1.12) 


Higherorder Equations


One can also solve higherorder equations. For example, in the equation
>


 (2.1) 
two solutions are determined by the rational function solver DEtools[ratsols]. Reduction of order then produces a final answer in terms of integrals:
>


 (2.2) 
>


 (2.3) 
Maple does a quick test to determine if a particular ODE is the symmetric product of a secondorder equation. If so, then a solution can be determined from the solutions of the secondorder equation. For example, the equation
>


 (2.4) 
found in Kamke or Abramowitz and Stegun is the symmetric power of Airy's equation. As such, dsolve produces:
>


 (2.5) 
Similarly,
>


 (2.6) 
>


 (2.7) 
>


 (2.8) 
Combined with previous methods, we find that the equation
>


 (2.9) 
has a single answer determined from the Maple exponential solver DEtools[expsols], and then reduction of order reduces to a symmetric equation. This gives
>


 (2.10) 
>


 (2.11) 


Additional Improvements


Improvements to solving linear ODEs also naturally lead to improvements in solving other differential equations. For example, firstorder Riccati equations are typically converted to secondorder linear equations to build their solutions. Similarly, linear systems of firstorder equations are solved by building one or more higherorder scalar equations and by constructing the matrix solutions. A simple example is found by
>


 (3.1) 
>


 (3.2) 
In the above answer, the functions AiryAi(1, .. ) and AiryBi(1,...) represent the derivatives of the AiryAi and AiryBi functions, respectively.

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