Closed-form Solutions of Linear Differential Equations
The Maple dsolve command allows determination of closed-form solutions for linear differential equations.
The dsolve command converts a homogeneous second-order linear differential equation of the form
ode:=a⁡x⁢ⅆ2ⅆx2⁢y⁡x+b⁡x⁢ⅆⅆx⁢y⁡x+c⁡x⁢y⁡x=0 into one of the form
new_ode:=ⅆ2ⅆx2⁢y⁡x+I⁡x⁢y⁡x=0, 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:
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.
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