Internal problem ID [7986]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 406.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_dAlembert]
Solve \begin {gather*} \boxed {a \left (y^{\prime }\right )^{2}-y y^{\prime }-x=0} \end {gather*}
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 264
dsolve(a*diff(y(x),x)^2-y(x)*diff(y(x),x)-x = 0,y(x), singsol=all)
\begin{align*} \frac {c_{1} \left (-y \relax (x )+\sqrt {4 x a +y \relax (x )^{2}}\right )}{\sqrt {-\frac {2 \left (y \relax (x ) \sqrt {4 x a +y \relax (x )^{2}}-2 a^{2}-2 x a -y \relax (x )^{2}\right )}{a^{2}}}}+x -\frac {\left (-y \relax (x )+\sqrt {4 x a +y \relax (x )^{2}}\right ) \arcsinh \left (\frac {-y \relax (x )+\sqrt {4 x a +y \relax (x )^{2}}}{2 a}\right )}{\sqrt {-\frac {2 \left (y \relax (x ) \sqrt {4 x a +y \relax (x )^{2}}-2 a^{2}-2 x a -y \relax (x )^{2}\right )}{a^{2}}}} = 0 \\ \frac {c_{1} \left (y \relax (x )+\sqrt {4 x a +y \relax (x )^{2}}\right )}{\sqrt {\frac {y \relax (x ) \sqrt {4 x a +y \relax (x )^{2}}+2 a^{2}+2 x a +y \relax (x )^{2}}{a^{2}}}}+x -\frac {\left (y \relax (x )+\sqrt {4 x a +y \relax (x )^{2}}\right ) \sqrt {2}\, \arcsinh \left (\frac {y \relax (x )+\sqrt {4 x a +y \relax (x )^{2}}}{2 a}\right )}{2 \sqrt {\frac {y \relax (x ) \sqrt {4 x a +y \relax (x )^{2}}+2 a^{2}+2 x a +y \relax (x )^{2}}{a^{2}}}} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.332 (sec). Leaf size: 71
DSolve[-x - y[x]*y'[x] + a*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\left \{x=-\frac {a K[1] \log \left (\sqrt {K[1]^2+1}-K[1]\right )}{\sqrt {K[1]^2+1}}+\frac {c_1 K[1]}{\sqrt {K[1]^2+1}},y(x)=a K[1]-\frac {x}{K[1]}\right \},\{y(x),K[1]\}\right ] \]