dsolve/formal_solution
find formal solutions to a homogeneous linear ODE with polynomial coefficients
Calling Sequence
Parameters
Description
Examples
dsolve(ODE, y(x), 'formal_solution', 'coeffs'=coeff_type, 'point'=x0)
dsolve(ODE, y(x), 'type=formal_solution', 'coeffs'=coeff_type, 'point'=x0)
ODE
-
homogeneous linear ordinary differential equation with polynomial coefficients
y(x)
dependent variable (the indeterminate function)
'type=formal_solution'
(optional) request for formal solutions
'coeffs'=coeff_type
(optional) coeff_type is one of 'mhypergeom', 'dAlembertian'
'point'=x0
algebraic number, rational in parameters, or infinity
When the input ODE is a homogeneous linear ode with polynomial coefficients, and the optional arguments 'formal_solution' (or 'type=formal_solution') and 'coeffs'=coeff_type are given, the dsolve command returns a set of formal solutions with the specified coefficients at the given point (the default is at the origin). For more information, see Slode[mhypergeom_formal_sol] and Slode[dAlembertian_formal_sol].
Find the formal solution set with m-hypergeometric series coefficients.
ode≔x2+1xdiffyx,x,x,x+32x2+1diffyx,x,x−12yx
ode≔x2+1xⅆ3ⅆx3yx+32x2+1ⅆ2ⅆx2yx−12yx
dsolveode,yx,formal_solution,coeffs=mhypergeom
yx=2x3+x_C1+_C2∑_n=1∞Γ_n−32−1_nx2_nΓ_n2πx
Find the formal solution set with d'Alembertian series coefficient.
ode≔−4−x2+2xyx+2x−3x3−x2diffyx,x+x3−x4diffyx,`$`x,2
ode≔−x2+2x−4yx+−3x3−x2+2xⅆⅆxyx+−x4+x3ⅆ2ⅆx2yx
dsolveode,yx,formal_solution,coeffs=dAlembertian
yx=x2−∑_n=0∞x_n2+∑_n=0∞∑_n1=0_n−1−12_n1∏_k=0_n1−1_k+2_k+32x_n_C1+ⅇ2x∑_n=0∞x_n−13_C2x
ode≔−x−1yx−2x2−4x−1diffyx,x−12xx+1x−6diffyx,`$`x,2+122+xx2diffyx,`$`x,3
ode≔−x−1yx−2x2−4x−1ⅆⅆxyx−xx+1x−6ⅆ2ⅆx2yx2+x+2x2ⅆ3ⅆx3yx2
dsolveode,yx,formal_solution,coeffs=dAlembertian,point=a
yx=_C1∑_n=0∞−1a+2_nx−a_n∏_k=0_n−1_k+1_k+2+_C2−a−2∑_n=0∞−1a+2_n∑_n1=0_n−1_n1+1∏_k=0_n1−1_k+1_k+22a+2a_n1_n1+2x−a_n∏_k=0_n−1_k+1_k+2+3aa+2∑_n=0∞−1a+2_n∑_n1=0_n−1_n1+1∏_k=0_n1−1_k+1_k+22a+2a_n1∑_n2=0_n1−1_n2+2∏_k=0_n2−1_k+2_k+3_n2+1_n1+2x−a_n∏_k=0_n−1_k+1_k+2−a2a+2∑_n=0∞−1a+2_n∑_n1=0_n−1_n1+1∏_k=0_n1−1_k+1_k+22a+2a_n1∑_n2=0_n1−1_n2+2∏_k=0_n2−1_k+2_k+3∑_n3=0_n2−1_n3+3−a_n3_n3+1∏_k=0_n3−1_k+4_k+1_k+3_n2+1_n1+2x−a_n∏_k=0_n−1_k+1_k+2_C3
See Also
DEtools/formal_sol
dsolve
dsolve/formal_series
Slode/dAlembertian_formal_sol
Slode/mhypergeom_formal_sol
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