Internal problem ID [15115]
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: 227.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _rational, _Clairaut]
\[ \boxed {x {y^{\prime }}^{2}-y y^{\prime }-y^{\prime }=-1} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 38
dsolve(x*diff(y(x),x)^2-y(x)*diff(y(x),x)-diff(y(x),x)+1=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -1-2 \sqrt {x} \\ y \left (x \right ) &= -1+2 \sqrt {x} \\ y \left (x \right ) &= \frac {c_{1}^{2} x -c_{1} +1}{c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.015 (sec). Leaf size: 46
DSolve[x*y'[x]^2-y[x]*y'[x]-y'[x]+1==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 x-1+\frac {1}{c_1} \\ y(x)\to \text {Indeterminate} \\ y(x)\to -2 \sqrt {x}-1 \\ y(x)\to 2 \sqrt {x}-1 \\ \end{align*}