dsolve/formal_solution - Maple Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Differential Equations : dsolve : dsolve/formal_solution

dsolve/formal_solution

find formal solutions to a homogeneous linear ODE with polynomial coefficients

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

dsolve(ODE, y(x), 'formal_solution', 'coeffs'=coeff_type, 'point'=x0)

dsolve(ODE, y(x), 'type=formal_solution', 'coeffs'=coeff_type, 'point'=x0)

Parameters

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

Description

• 

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].

Examples

Find the formal solution set with m-hypergeometric series coefficients.

odex2+1xⅆ3ⅆx3yx+32x2+1ⅆ2ⅆx2yx12yx

ode:=x2+1xⅆ3ⅆx3yx+32x2+1ⅆ2ⅆx2yx12yx

(1)

dsolveode,yx,'formal_solution','coeffs'='mhypergeom'

yx=2x3+x_C1+12_C2_n=1∞Γ_n321_nx2_nΓ_nπx

(2)

Find the formal solution set with d'Alembertian series coefficient.

ode4x2+2xyx+2x3x3x2ⅆⅆxyx+x3x4ⅆ2ⅆx2yx

ode:=x2+2x4yx+3x3x2+2xⅆⅆxyx+x4+x3ⅆ2ⅆx2yx

(3)

dsolveode,yx,'formal_solution','coeffs'='dAlembertian'

yx=x212_n=0∞x_n+_n=0∞_n1=0_n112_n1_k=0_n11_k+32_k+2x_n_C1+ⅇ2x_n=0∞x_n13_C2x

(4)

odex1yx2x24x1ⅆⅆxyx1xx+1x6ⅆ2ⅆx2yx2+12+xx2ⅆ3ⅆx3yx2

ode:=x1yx2x24x1ⅆⅆxyx12xx+1x6ⅆ2ⅆx2yx+12x+2x2ⅆ3ⅆx3yx

(5)

dsolveode,yx,'formal_solution','coeffs'='dAlembertian','point'=a

yx=_C1_n=0∞1a+2_n_k=0_n1_k+2_k+1xa_n+_C2a2_n=0∞1a+2_n_k=0_n1_k+2_k+1_n1=0_n1_n1+1a+2a_n1_k=0_n11_k+12_k+22_n1+2xa_n+3aa+2_n=0∞1a+2_n_k=0_n1_k+2_k+1_n1=0_n1_n1+1a+2a_n1_k=0_n11_k+12_k+22_n2=0_n11_n2+2_k=0_n21_k+2_k+3_n2+1_n1+2xa_na2a+2_n=0∞1a+2_n_k=0_n1_k+2_k+1_n1=0_n1_n1+1a+2a_n1_k=0_n11_k+12_k+22_n2=0_n11_n2+2_k=0_n21_k+2_k+3_n3=0_n21_n3+3a_n3_k=0_n31_k+3_k+4_k+1_n3+1_n2+1_n1+2xa_n_C3

(6)

See Also

DEtools/formal_sol

dsolve

dsolve/formal_series

Slode/dAlembertian_formal_sol

Slode/mhypergeom_formal_sol

 


Download Help Document

Was this information helpful?



Please add your Comment (Optional)
E-mail Address (Optional)
What is ? This question helps us to combat spam