Internal problem ID [8113]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 533.
ODE order: 1.
ODE degree: 3.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _Clairaut]
Solve \begin {gather*} \boxed {x \left (y^{\prime }\right )^{3}-y \left (y^{\prime }\right )^{2}+a=0} \end {gather*}
✓ Solution by Maple
Time used: 0.156 (sec). Leaf size: 92
dsolve(x*diff(y(x),x)^3-y(x)*diff(y(x),x)^2+a=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {3 \,2^{\frac {1}{3}} \left (x^{2} a \right )^{\frac {1}{3}}}{2} \\ y \relax (x ) = -\frac {3 \,2^{\frac {1}{3}} \left (x^{2} a \right )^{\frac {1}{3}}}{4}-\frac {3 i \sqrt {3}\, 2^{\frac {1}{3}} \left (x^{2} a \right )^{\frac {1}{3}}}{4} \\ y \relax (x ) = -\frac {3 \,2^{\frac {1}{3}} \left (x^{2} a \right )^{\frac {1}{3}}}{4}+\frac {3 i \sqrt {3}\, 2^{\frac {1}{3}} \left (x^{2} a \right )^{\frac {1}{3}}}{4} \\ y \relax (x ) = x c_{1}+\frac {a}{c_{1}^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.013 (sec). Leaf size: 89
DSolve[a - y[x]*y'[x]^2 + x*y'[x]^3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {a}{c_1{}^2}+c_1 x \\ y(x)\to \frac {3 \sqrt [3]{a} x^{2/3}}{2^{2/3}} \\ y(x)\to -\frac {3 \sqrt [3]{-1} \sqrt [3]{a} x^{2/3}}{2^{2/3}} \\ y(x)\to \frac {3 (-1)^{2/3} \sqrt [3]{a} x^{2/3}}{2^{2/3}} \\ \end{align*}