Solving d'Alembert ODEs - Maple Help

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Solving d'Alembert ODEs

Description

• 

The general form of the d'Alembert ODE is given by:

dAlembert_ode := y(x)=x*f(diff(y(x),x))+g(diff(y(x),x));

dAlembert_ode:=yx=xfⅆⅆxyx+gⅆⅆxyx

(1)
  

where f and g are arbitrary functions. See Differentialgleichungen, by E. Kamke, p. 31. This ODE is actually a generalization of the Clairaut ODE, and is almost always dealt with by looking for a solution in parametric form. For more information, see odeadvisor[patterns].

Examples

withDEtools,odeadvisor

odeadvisor

(2)

odeadvisordAlembert_ode

_dAlembert

(3)

The general form of the solution for the d'Alembert ODE is returned by dsolve in parametric form, together with a possible singular solution, as follows:

dsolvedAlembert_ode

yx=xRootOf_Zf_Z+gRootOf_Zf_Z,x_T=ⅇ∫ⅆⅆ_Tf_T_Tf_Tⅆ_T∫ⅆⅆ_Tg_Tⅇ∫ⅆⅆ_Tf_T_Tf_Tⅆ_T_Tf_Tⅆ_T+_C1,y_T=ⅇ∫ⅆⅆ_Tf_T_Tf_Tⅆ_T∫ⅆⅆ_Tg_Tⅇ∫ⅆⅆ_Tf_T_Tf_Tⅆ_T_Tf_Tⅆ_T+_C1f_T+g_T

(4)

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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