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Solving ODEs That Do Not Contain Either the Dependent or Independent Variable

 

Description

Examples

Description

• 

The general form of an nth order ODE that is missing the dependent variable is:

missing_y_ode := F(x,'seq(diff(y(x),x$i),i=1..n)');

(1)
  

where F is an arbitrary function of its arguments. The order can be reduced by introducing a new variable p(x) = diff(y(x),x). If the reduced ODE can be solved for p(x), the solution to the original ODE is determined as a quadrature.

• 

The general form of an nth order ODE that is missing the independent variable is:

missing_x_ode := F(y(x),'seq(diff(y(x),x$i),i=1..n)');

(2)
  

where F is an arbitrary function of its arguments. The transformation

  

yields a reduction of order. If the reduced ODE can be solved for p(y), the solution to the original ODE can be given implicitly as

x = Int(1/p(y),y) + _C1;

(3)
  

See Murphy, "Ordinary Differential Equations and their Solutions", 1960, sections B2(1,2), and C2(1,2).

Examples

(4)

(5)

(6)

(7)

Explicit and implicit answers can be tested, in principle, using odetest:

(8)

(9)

(10)

(11)

In the case of multiple answers it is convenient to "map" odetest as follows:

(12)

The most general third order ODE missing x. This ODE cannot be solved to the end: its solution involves the solving of the most general second order ODE. However, its differential order can be reduced (see ?dsolve,ODESolStruc):

(13)

(14)

(15)

(16)

The most general third order ODE missing y.

(17)

(18)

(19)

(20)

See Also

DEtools

odeadvisor

dsolve

quadrature

missing

reducible

linear_ODEs

exact_linear

exact_nonlinear

odeadvisor,types

 


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