Internal problem ID [7984]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 405.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_dAlembert]
\[ \boxed {a {y^{\prime }}^{2}+y y^{\prime }-x=0} \]
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 379
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 \left (x \right )+\sqrt {4 a x +y \left (x \right )^{2}}\right )}{\sqrt {\frac {-y \left (x \right )+\sqrt {4 a x +y \left (x \right )^{2}}-2 a}{a}}\, \sqrt {\frac {-y \left (x \right )+\sqrt {4 a x +y \left (x \right )^{2}}+2 a}{a}}}+x +\frac {\left (-y \left (x \right )+\sqrt {4 a x +y \left (x \right )^{2}}\right ) \ln \left (\frac {\sqrt {\frac {4 a x +2 y \left (x \right )^{2}-2 y \left (x \right ) \sqrt {4 a x +y \left (x \right )^{2}}-4 a^{2}}{a^{2}}}\, a +\sqrt {4 a x +y \left (x \right )^{2}}-y \left (x \right )}{2 a}\right )}{\sqrt {-\frac {2 \left (y \left (x \right ) \sqrt {4 a x +y \left (x \right )^{2}}+2 a^{2}-2 a x -y \left (x \right )^{2}\right )}{a^{2}}}} = 0 \\ \frac {c_{1} \left (y \left (x \right )+\sqrt {4 a x +y \left (x \right )^{2}}\right )}{\sqrt {\frac {-2 y \left (x \right )-2 \sqrt {4 a x +y \left (x \right )^{2}}-4 a}{a}}\, \sqrt {\frac {-2 y \left (x \right )-2 \sqrt {4 a x +y \left (x \right )^{2}}+4 a}{a}}}+x -\frac {\left (y \left (x \right )+\sqrt {4 a x +y \left (x \right )^{2}}\right ) \sqrt {2}\, \ln \left (\frac {\sqrt {2}\, \sqrt {\frac {y \left (x \right ) \sqrt {4 a x +y \left (x \right )^{2}}-2 a^{2}+2 a x +y \left (x \right )^{2}}{a^{2}}}\, a -\sqrt {4 a x +y \left (x \right )^{2}}-y \left (x \right )}{2 a}\right )}{2 \sqrt {\frac {y \left (x \right ) \sqrt {4 a x +y \left (x \right )^{2}}-2 a^{2}+2 a x +y \left (x \right )^{2}}{a^{2}}}} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.582 (sec). Leaf size: 79
DSolve[-x + y[x]*y'[x] + a*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\left \{x=-\frac {2 a K[1] \arctan \left (\frac {\sqrt {1-K[1]^2}}{K[1]+1}\right )}{\sqrt {1-K[1]^2}}+\frac {c_1 K[1]}{\sqrt {1-K[1]^2}},y(x)=\frac {x}{K[1]}-a K[1]\right \},\{y(x),K[1]\}\right ] \]