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));
 ${\mathrm{Clairaut_ode}}{:=}{y}{}\left({x}\right){=}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right){+}{g}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right)$ (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);
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{x}{+}{g}{}\left({\mathrm{_C1}}\right)$ (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

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{odeadvisor}\right)$
 $\left[{\mathrm{odeadvisor}}\right]$ (3)
 > $\mathrm{odeadvisor}\left(\mathrm{Clairaut_ode}\right)$
 $\left[{\mathrm{_Clairaut}}\right]$ (4)
 > $\mathrm{ode}:=y\left(x\right)=x\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)+\mathrm{cos}\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)$
 ${\mathrm{ode}}{:=}{y}{}\left({x}\right){=}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right){+}{\mathrm{cos}}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right)$ (5)
 > $\mathrm{ans}:=\mathrm{dsolve}\left(\mathrm{ode}\right)$
 ${\mathrm{ans}}{:=}{y}{}\left({x}\right){=}{\mathrm{arcsin}}{}\left({x}\right){}{x}{+}\sqrt{{-}{{x}}^{{2}}{+}{1}}{,}{y}{}\left({x}\right){=}{\mathrm{_C1}}{}{x}{+}{\mathrm{cos}}{}\left({\mathrm{_C1}}\right)$ (6)

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

 See Also DEtools, odeadvisor, dsolve, and ?odeadvisor, where 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.