Internal problem ID [2345]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 38, page 173
Problem number: 8.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_dAlembert]
\[ \boxed {-y^{\prime } y-{y^{\prime }}^{2}=-x} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 300
dsolve(x=diff(y(x),x)*y(x)+diff(y(x),x)^2,y(x), singsol=all)
\begin{align*} \frac {\left (-y \left (x \right )+\sqrt {y \left (x \right )^{2}+4 x}\right ) c_{1}}{\sqrt {-2 y \left (x \right )+2 \sqrt {y \left (x \right )^{2}+4 x}-4}\, \sqrt {-2 y \left (x \right )+2 \sqrt {y \left (x \right )^{2}+4 x}+4}}+x +\frac {\left (-y \left (x \right )+\sqrt {y \left (x \right )^{2}+4 x}\right ) \ln \left (-\frac {y \left (x \right )}{2}+\frac {\sqrt {y \left (x \right )^{2}+4 x}}{2}+\frac {\sqrt {2 y \left (x \right )^{2}-2 y \left (x \right ) \sqrt {y \left (x \right )^{2}+4 x}+4 x -4}}{2}\right )}{\sqrt {2 y \left (x \right )^{2}-2 y \left (x \right ) \sqrt {y \left (x \right )^{2}+4 x}+4 x -4}} = 0 \frac {\left (y \left (x \right )+\sqrt {y \left (x \right )^{2}+4 x}\right ) c_{1}}{\sqrt {-2 y \left (x \right )-2 \sqrt {y \left (x \right )^{2}+4 x}-4}\, \sqrt {-2 y \left (x \right )-2 \sqrt {y \left (x \right )^{2}+4 x}+4}}+x -\frac {\left (y \left (x \right )+\sqrt {y \left (x \right )^{2}+4 x}\right ) \ln \left (-\frac {y \left (x \right )}{2}-\frac {\sqrt {y \left (x \right )^{2}+4 x}}{2}+\frac {\sqrt {2 y \left (x \right )^{2}+2 y \left (x \right ) \sqrt {y \left (x \right )^{2}+4 x}+4 x -4}}{2}\right )}{\sqrt {2 y \left (x \right )^{2}+2 y \left (x \right ) \sqrt {y \left (x \right )^{2}+4 x}+4 x -4}} = 0 \end{align*}
✓ Solution by Mathematica
Time used: 0.497 (sec). Leaf size: 77
DSolve[x==y'[x]*y[x]+y'[x]^2,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\left \{x=-\frac {2 K[1] \arctan \left (\frac {\sqrt {1-K[1]^2}}{K[1]+1}\right )}{\sqrt {1-K[1]^2}}+\frac {c_1 K[1]}{\sqrt {1-K[1]^2}},y(x)=\frac {x}{K[1]}-K[1]\right \},\{y(x),K[1]\}\right ] \]