Closed-form Solutions of Linear Differential Equations
The Maple dsolve command allows determination of closed-form solutions for linear differential equations.
Second-order Equations
The dsolve command converts a homogeneous second-order linear differential equation of the form
ode:=axⅆ2ⅆx2yx+bxⅆⅆxyx+cxyx=0 into one of the form
new_ode:=ⅆ2ⅆx2yx+Ixyx=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):
restart
ode1:=ⅆ2ⅆx2yx+2x2ⅆⅆxyx+2xyx
ode1≔ⅆ2ⅆx2yx+2x2ⅆⅆxyx+2xyx
ans1:=dsolveode1,yx
ans1≔yx=c__1ⅇ−x33xBesselI16,x33+c__2ⅇ−x33xBesselK16,x33
ode2:=4x4−1ⅆⅆxⅆⅆxyx+12x4+1ⅆⅆxyxx+16x2−u24x4−1−vv+1yx
ode2≔4x4−1ⅆ2ⅆx2yx+12x4+1ⅆⅆxyxx+16x2−u24x4−1−vv+1yx
ans2:=dsolveode2,yx
ans2≔yx=c__1LegendrePv,u,2x2+c__2LegendreQv,u,2x2
ode3:=ⅆ2ⅆx2yx−xⅆⅆxyx+nyx
ode3≔ⅆ2ⅆx2yx−xⅆⅆxyx+nyx
ans3:=dsolveode3,yx
ans3≔yx=c__1xKummerM12−n2,32,x22+c__2xKummerU12−n2,32,x22
ode4:=x−9x5ⅆ2ⅆx2yx+1−9x4ⅆⅆxyx+36x3yx
ode4≔−9x5+xⅆ2ⅆx2yx+−9x4+1ⅆⅆxyx+36x3yx
ans4:=dsolveode4,yx
ans4≔yx=c__1EllipticE3x2+c__2EllipticCE3x2−EllipticCK3x2
In all of these examples, one can verify the solutions by using the odetest command:
odetestans1,ode1
0
odetestans2,ode2
odetestans3,ode3
odetestans4,ode4
Higher-order Equations
One can also solve higher-order equations. For example, in the equation
ode:=−300yx+192xⅆⅆxyx+−47x2+4x3μ−x4−4x2ν2ⅆ2ⅆx2yx+12yxx2−48yxμx+48yxν2+4x4ⅆ4ⅆx4yx
ode≔−300yx+192xⅆⅆxyx+4x3μ−4x2ν2−x4−47x2ⅆ2ⅆx2yx+12yxx2−48yxμx+48yxν2+4x4ⅆ4ⅆx4yx
two solutions are determined by the rational function solver DEtools[ratsols]. Reduction of order then produces a final answer in terms of integrals:
dsolveode,yx
yx=c__1x3+c__2x4+c__3−∫WhittakerMμ,ν,xx3ⅆxx7+∫x4WhittakerMμ,ν,xⅆxx3+c__4∫WhittakerWμ,ν,xx3ⅆxx7−∫x4WhittakerWμ,ν,xⅆxx3
odetest,ode
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
ode:=ⅆ3ⅆx3yx−4xⅆⅆxyx−2yx
ode≔ⅆ3ⅆx3yx−4xⅆⅆxyx−2yx
found in Kamke or Abramowitz and Stegun is the symmetric power of Airy's equation. As such, dsolve produces:
yx=c__1AiryAix2+c__2AiryBix2+c__3AiryAixAiryBix
Similarly,
ode2:=ⅆ3ⅆx3yx−6xⅆ2ⅆx2yx+24x2+2a−1ⅆⅆxyx−8axyx
ode2≔ⅆ3ⅆx3yx−6ⅆ2ⅆx2yxx+24x2+2a−1ⅆⅆxyx−8axyx
ans2≔yx=c__1x2KummerM12−a4,32,x22+c__2x2KummerU12−a4,32,x22+c__3x2KummerM12−a4,32,x2KummerU12−a4,32,x2
Combined with previous methods, we find that the equation
ode:=2yx+−2+4xⅆⅆxyx−4xⅆ2ⅆx2yx−ⅆ3ⅆx3yx+ⅆ4ⅆx4yx
ode≔2yx+−2+4xⅆⅆxyx−4ⅆ2ⅆx2yxx−ⅆ3ⅆx3yx+ⅆ4ⅆx4yx
has a single answer determined from the Maple exponential solver DEtools[expsols], and then reduction of order reduces to a symmetric equation. This gives
yx=c__1ⅇx+c__2∫AiryAix2ⅇ−xⅆxⅇx+c__3∫AiryBix2ⅇ−xⅆxⅇx+c__4∫AiryAixAiryBixⅇ−xⅆxⅇx
Additional Improvements
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
ode:=ⅆⅆxyx=x+yx+ayx2
ode≔ⅆⅆxyx=x+yx+ayx2
yx=2AiryAi1,−4ax−14a23a13c__1−AiryAi−4ax−14a23c__1+2AiryBi1,−4ax−14a23a13−AiryBi−4ax−14a232aAiryAi−4ax−14a23c__1+AiryBi−4ax−14a23
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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