Solving Clairaut ODEs
Description
Examples
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
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
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.
withDEtools,odeadvisor
odeadvisor
odeadvisorClairaut_ode
_Clairaut
ode≔yx=xdiffyx,x+cosdiffyx,x
ode≔yx=xⅆⅆxyx+cosⅆⅆxyx
ans≔dsolveode
ans≔yx=arcsinxx+−x2+1,yx=c__1x+cosc__1
Note the absence of integration constant _C in the singular solution present in the above.
See Also
DEtools
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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