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dsolve

Find formal power series solutions to a linear ODE with polynomial coefficients

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

dsolve(ODE, y(x), 'formal_series', 'coeffs'=coeff_type)

dsolve(ODE, y(x), 'type=formal_series', 'coeffs'=coeff_type)

Parameters

ODE

-

linear ordinary differential equation with polynomial coefficients

y(x)

-

the dependent variable (the indeterminate function)

'type=formal_series'

-

request for formal power series solutions

'coeffs'=coeff_type

-

coeff_type is one of 'polynomial', 'rational', 'hypergeom', 'mhypergeom'

Description

• 

When the input ODE is a linear ode with polynomial coefficients which is homogeneous or inhomogeneous with rational right hand side, and the optional arguments 'formal_series' (or 'type=formal_series') and 'coeffs'=coeff_type are given, dsolve will return a set of formal power series solutions with the specified coefficients at all candidate points of expansion. See Slode for more details.

Examples

Formal power series solution with polynomial coefficients

ode := (3*x^2-6*x+3)*diff(diff(y(x),x),x) + (12*x-12)*diff(y(x),x)+6*y(x);

ode3x26x+3ⅆ2ⅆx2yx+12x12ⅆⅆxyx+6yx

(1)

dsolve(ode,y(x),'formal_series','coeffs'='polynomial');

yx=_n=0_C2_n+_C1x_n

(2)

Formal power series solution with rational coefficients

ode := (3-x)*diff(diff(y(x),x),x)-diff(y(x),x);

ode3xⅆ2ⅆx2yxⅆⅆxyx

(3)

dsolve(ode,y(x),'formal_series','coeffs'='rational');

yx=_C2+_C1_n=1x2_n_n

(4)

Formal power series solution with hypergeometric coefficients

ode := 2*x*(x-1)*diff(diff(y(x),x),x)+(7*x-3)*diff(y(x),x)+2*y(x) = 0;

ode2xx1ⅆ2ⅆx2yx+7x3ⅆⅆxyx+2yx=0

(5)

dsolve(ode,y(x),'type=formal_series','coeffs'='hypergeom');

yx=_C1_n=0_n+1x_n2_n+1,yx=_C1_n=0Γ12+_nx+1_nπ_n!,yx=_C1_n=0Γ12+_n−1_nx1_nΓ_n+1π

(6)

Formal m-sparse m-hypergeometric power series solutions

ode := diff(y(x),x,x)+(x-1)*y(x);

odeⅆ2ⅆx2yx+x1yx

(7)

dsolve(ode,y(x),'type=formal_series','coeffs'='mhypergeom');

yx=_C1Γ23_n=019_nx13_nΓ_n+1Γ_n+23,yx=2_C1π3_n=019_nx13_n+1Γ_n+43Γ_n+19Γ23

(8)

See Also

dsolve

dsolve,formal_solution

Slode