Internal problem ID [15113]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Section 8.3. The Lagrange and Clairaut equations. Exercises page 72
Problem number: 225.
ODE order: 1.
ODE degree: 3.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _Clairaut]
\[ \boxed {y-x y^{\prime }-\frac {a}{{y^{\prime }}^{2}}=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 76
dsolve(y(x)=x*diff(y(x),x)+a/diff(y(x),x)^2,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}} \left (1+i \sqrt {3}\right )}{4} \\ y \left (x \right ) &= \frac {3 \,2^{\frac {1}{3}} \left (a \,x^{2}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{4} \\ y \left (x \right ) &= \frac {c_{1}^{3} x +a}{c_{1}^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.012 (sec). Leaf size: 89
DSolve[y[x]==x*y'[x]+a/y'[x]^2,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*}