Solving ODEs That Are in Quadrature Format - Maple Help

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Solving ODEs That Are in Quadrature Format

Description

• 

An ODE is said to be in quadrature format when the following conditions are met:

  

1) the ODE is of first order and the right hand sides below depend only on x or y(x):

quadrature_1_x_ode := diff(y(x),x)=F(x);

quadrature_1_x_ode:=ⅆⅆxyx=Fx

(1)

quadrature_1_y_ode := diff(y(x),x)=F(y(x));

quadrature_1_y_ode:=ⅆⅆxyx=Fyx

(2)
  

2) the ODE is of high order and the right hand side depends only on x. For example:

quadrature_h_x_ode := diff(y(x),x,x,x,x)=F(x);

quadrature_h_x_ode:=ⅆ4ⅆx4yx=Fx

(3)
  

where F is an arbitrary function. These ODEs are just integrals in disguised format, and are solved mainly by integrating both sides.

Examples

withDEtools,odeadvisor,symgen

odeadvisor,symgen

(4)

odeadvisorquadrature_1_x_ode

_quadrature

(5)

dsolvequadrature_1_x_ode

yx=∫Fxⅆx+_C1

(6)

odeadvisorquadrature_1_y_ode

_quadrature

(7)

dsolvequadrature_1_y_ode

x∫yx1F_aⅆ_a+_C1=0

(8)

From the point of view of their symmetries, all ODEs "missing y" have the symmetry [xi = 0, eta = 1], and all ODEs "missing x" have the symmetry [xi = 1, eta = 0] (see symgen);

symgenquadrature_1_x_ode

_ξ=0,_η=1

(9)

symgenquadrature_1_y_ode

_ξ=1,_η=0

(10)

See Also

DEtools, odeadvisor, dsolve, and ?odeadvisor,<TYPE> where <TYPE> is one of: quadrature, linear, separable, Bernoulli, exact, homogeneous, homogeneousB, homogeneousC, homogeneousD, homogeneousG, Chini, Riccati, Abel, Abel2A, Abel2C, rational, Clairaut, dAlembert, sym_implicit, patterns; for other differential orders see odeadvisor,types.


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