Solving Clairaut ODEs - Maple Help

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Solving Clairaut ODEs

Description

• 

The general form of Clairaut's ODE is given by:

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

Clairaut_ode:=yx=xⅆⅆxyx+gⅆⅆxyx

(1)
  

where g is an arbitrary function of dy/dx. See Differentialgleichungen, by E. Kamke, p. 31. This type of equation always has a linear solution:

y(x) = _C1*x + g(_C1);

yx=_C1x+g_C1

(2)
• 

It is also worth mentioning that singular nonlinear solutions can be obtained by looking for a solution in parametric form. For more information, see odeadvisor/parametric.

Examples

withDEtools,odeadvisor

odeadvisor

(3)

odeadvisorClairaut_ode

_Clairaut

(4)

ode:=yx=xⅆⅆxyx+cosⅆⅆxyx

ode:=yx=xⅆⅆxyx+cosⅆⅆxyx

(5)

ans:=dsolveode

ans:=yx=arcsinxx+x2+1,yx=_C1x+cos_C1

(6)

Note the absence of integration constant _C in the singular solution present in the above.

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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