The dsolve command converts a homogeneous second-order 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):

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

One can also solve higher-order equations. For example, in the equation

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

Maple does a quick test to determine if a particular ODE is the symmetric product of a second-order equation. If so, then a solution can be determined from the solutions of the second-order equation. For example, the equation

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

Similarly,

Combined with previous methods, we find that the equation

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

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

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