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Solving ODEs That Are in Quadrature Format

 

Description

Examples

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);

(1)

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

(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);

(3)
  

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

Examples

(4)

(5)

(6)

(7)

(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);

(9)

(10)

See Also

DEtools

odeadvisor

dsolve

quadrature

linear

separable

Bernoulli

exact

homogeneous

homogeneousB

homogeneousC

homogeneousD

homogeneousG

Chini

Riccati

Abel

Abel2A

Abel2C

rational

Clairaut

dAlembert

sym_implicit

patterns

odeadvisor,types

 


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