Internal problem ID [4279]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 35
Problem number: 1058.
ODE order: 1.
ODE degree: 3.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _Clairaut]
\[ \boxed {x {y^{\prime }}^{3}-y {y^{\prime }}^{2}=-a} \]
✓ Solution by Maple
Time used: 0.328 (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 \left (x \right ) = \frac {3 \,2^{\frac {1}{3}} \left (a \,x^{2}\right )^{\frac {1}{3}}}{2} y \left (x \right ) = -\frac {3 \,2^{\frac {1}{3}} \left (a \,x^{2}\right )^{\frac {1}{3}}}{4}-\frac {3 i \sqrt {3}\, 2^{\frac {1}{3}} \left (a \,x^{2}\right )^{\frac {1}{3}}}{4} y \left (x \right ) = -\frac {3 \,2^{\frac {1}{3}} \left (a \,x^{2}\right )^{\frac {1}{3}}}{4}+\frac {3 i \sqrt {3}\, 2^{\frac {1}{3}} \left (a \,x^{2}\right )^{\frac {1}{3}}}{4} y \left (x \right ) = c_{1} x +\frac {a}{c_{1}^{2}} \end{align*}
✓ Solution by Mathematica
Time used: 0.015 (sec). Leaf size: 89
DSolve[x (y'[x])^3 - y[x] (y'[x])^2 +a==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*}