Internal problem ID [6064]
Book: Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam
Publishing Co. NY. 6th edition. 1981.
Section: CHAPTER 16. Nonlinear equations. Section 99. Clairaut’s equation. EXERCISES Page
320
Problem number: 26.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_rational, _dAlembert]
Solve \begin {gather*} \boxed {2 x \left (y^{\prime }\right )^{2}+\left (-y+2 x \right ) y^{\prime }+1-y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.297 (sec). Leaf size: 146
dsolve(2*x*diff(y(x),x)^2+(2*x-y(x))*diff(y(x),x)+1-y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = -\left (-2 \left ({\mathrm e}^{\RootOf \left (-x \,{\mathrm e}^{3 \textit {\_Z}}+2 x \,{\mathrm e}^{2 \textit {\_Z}}+c_{1} {\mathrm e}^{\textit {\_Z}}+\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{\textit {\_Z}} x +1\right )}-1\right )^{2}-2 \,{\mathrm e}^{\RootOf \left (-x \,{\mathrm e}^{3 \textit {\_Z}}+2 x \,{\mathrm e}^{2 \textit {\_Z}}+c_{1} {\mathrm e}^{\textit {\_Z}}+\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{\textit {\_Z}} x +1\right )}+2\right ) {\mathrm e}^{-\RootOf \left (-x \,{\mathrm e}^{3 \textit {\_Z}}+2 x \,{\mathrm e}^{2 \textit {\_Z}}+c_{1} {\mathrm e}^{\textit {\_Z}}+\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{\textit {\_Z}} x +1\right )} x +{\mathrm e}^{-\RootOf \left (-x \,{\mathrm e}^{3 \textit {\_Z}}+2 x \,{\mathrm e}^{2 \textit {\_Z}}+c_{1} {\mathrm e}^{\textit {\_Z}}+\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{\textit {\_Z}} x +1\right )} \]
✓ Solution by Mathematica
Time used: 2.291 (sec). Leaf size: 49
DSolve[2*x*(y'[x])^2+(2*x-y[x])*y'[x]+1-y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\left \{x=\frac {\frac {1}{K[1]+1}+\log (K[1]+1)}{K[1]^2}+\frac {c_1}{K[1]^2},y(x)=2 x K[1]+\frac {1}{K[1]+1}\right \},\{y(x),K[1]\}\right ] \]