gfun
algeqtodiffeq
compute a differential equation satisfied by an algebraic function
Calling Sequence
Parameters
Description
Examples
algeqtodiffeq(p, y(z), ini)
p
-
polynomial in y and z (or a polynomial equation)
y
name; holonomic function name
z
name; variable of the holonomic function y
ini
(optional) set; specify computation of initial conditions for the resulting differential equation
The algeqtodiffeq(p, y(z)) command computes a linear differential equation with polynomial coefficients verified by the function y(z). The polynomial p defines an algebraic function, RootOfp,y in Maple terms. The resulting equation is of order at most degreep,y−1.
The resulting linear differential equation contains initial conditions in zero (y0, Dy0, and so on), and can thus be passed directly to dsolve. In general, y0 is a RootOf a polynomial, Dy0 a rational expression in y0, D2y0 a rational expression in y0, Dy0, and so on.
If initial conditions are specified using ini, the algeqtodiffeq function attempts to compute initial conditions for the resulting differential equation.
If a particular solution of eq is selected by specifying initial terms of its power series expansion at the origin using the same syntax as that of initial conditions for dsolve, the algeqtodiffeq function returns the corresponding initial conditions together with the differential equation.
withgfun:
algeqtodiffeqy=1+zy2,yz
1+−1+2zyz+4z2−zⅆⅆzyz
algeqtodiffeq56a3+7a3y3−14yz,yz,y0=−2
−yzz+3ⅆⅆzyzz2+−108a9+2z3ⅆ2ⅆz2yz,y0=−2,Dy0=−13a3
We can use algeqtodiffeq with diffeqtorec to determine fast Taylor expansions.
p≔y=1+zy+zy5
p≔y=zy5+zy+1
deq≔algeqtodiffeqp,yz
deq≔−147840z3+169920z2−22320z+240yz+393216z8−294912z7−466944z6+92160z5+447360z4+65664z3−265488z2+29064z−120ⅆⅆzyz+589824z9−1130496z8−30720z7+878592z6−132480z5−444384z4+203256z3+67800z2−1392zⅆ2ⅆz2yz+196608z10−606208z9+489472z8+165888z7+83200z6+626112z5+271848z4+24488z3−1408z2ⅆ3ⅆz3yz+16384z11−69632z10+105472z9−56064z8+188480z7+166896z6+38012z5+1333z4−256z3ⅆ4ⅆz4yz,y0=1
rec≔diffeqtorecdeq,yz,un
rec≔16384n4+98304n3+180224n2+98304nun+−69632n4−466944n3−1060864n2−958464n−294912un+1+105472n4+700416n3+1332224n2+208896n−995328un+2+−56064n4−170496n3+1257216n2+5973504n+6543360un+3+188480n4+1968000n3+7213120n2+11107200n+6572160un+4+166896n4+2962656n3+18918576n2+51195456n+49204800un+5+38012n4+956064n3+8804404n2+35087184n+50788512un+6+1333n4+53814n3+747191n2+4381134n+9313488un+7+−256n4−8064n3−95216n2−499464n−982080un+8,u0=1,u1=2,u2=12,u3=112,u4=1232,u5=14832,u6=189184,u7=2512064
p_generator:=rectoproc(rec,u(n),list):
p_generator30
1,2,12,112,1232,14832,189184,2512064,34358784,480745984,6848734464,99003237376,1448575666176,21411827808256,319255531155456,4796005997940736,72520546008219648,1102912584949792768,16859182461720526848,258886644574700699648,3991711460817459806208,61775021926688418365440,959229931916911121530880,14940323391360408046796800,233352506098550016631111680,3654109325605190169830359040,57356169042767373344673103872,902258550678887785413876908032,14222150544386213214581667397632,224605955315162319622246867402752,3553395680818726488887774467325952
See Also
dsolve
gfun/parameters
gfun[diffeqtorec]
gfun[rectoproc]
RootOf
Download Help Document