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DEtools[dperiodic_sols] - find doubly-periodic solutions of a linear ODE
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Calling Sequence
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dperiodic_sols(lode, v)
dperiodic_sols(coeff_list, x)
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Parameters
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lode
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linear ODE in diff form
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v
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dependent variable of lode
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coeff_list
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list of coefficients of a linear ODE; specified in order of increasing differential order
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x
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independent variable of a linear ODE
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Description
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The dperiodic_sols function seeks closed form solutions of linear ODEs having doubly-periodic coefficients. It returns either one or more doubly-periodic solutions to the linear ODE or provides a proof that no such solution exists.
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In the case of an order two linear ODE, the dperiodic_sols function also seeks a general solution in terms of solutions that are doubly-periodic or doubly-periodic of the second kind.
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The dperiodic_sols function returns, if possible, a list of one or more independent solutions. To find only doubly periodic solutions, set the environment variable _EnvDperiodicOnly to true.
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The dperiodic_sols(lode, v) and dperiodic_sols(coeff_list, x) calling sequences are equivalent.
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The dperiodic_sols(coeff_list, x) calling sequence is convenient for programming.
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Examples
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Special case of Lame equation (n=1).
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![[exp(-(1/2)*(-g3+4*B^3-g2*B)^(1/2)*(Int(1/(-P+B), x)))*(-P+B)^(1/2), exp((1/2)*(-g3+4*B^3-g2*B)^(1/2)*(Int(1/(-P+B), x)))*(-P+B)^(1/2)]](/support/helpjp/helpview.aspx?si=8687/file01322/math139.png)
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Kamke 2.74
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![[JacobiSN(x, 3)-(4/3)*JacobiSN(x, 3)^3, (JacobiSN(x, 3)-(4/3)*JacobiSN(x, 3)^3)*(18*JacobiCN(x, 3)*JacobiSN(x, 3)/(JacobiDN(x, 3)^2+23/4)+9*JacobiCN(x, 3)*JacobiSN(x, 3)/(-1+JacobiDN(x, 3)^2))]](/support/helpjp/helpview.aspx?si=8687/file01322/math156.png)
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Kamke 2.73 (Only doubly periodic solutions).
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![[P]](/support/helpjp/helpview.aspx?si=8687/file01322/math176.png)
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References
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Burger, R.; Labahn, G.; and van Hoeij, M. "Closed form solutions of linear odes having elliptic functions as coefficients." Proceedings of ISSAC'04, Santander, Spain, ACM Press, (2004): 58-64.
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