Internal problem ID [8742]
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]
\[ \boxed {a {y^{\prime }}^{2}-y y^{\prime }=x} \]
✓ 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 \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^{2}+2 a x +y \left (x \right )^{2}}{a^{2}}}}+x -\frac {\sqrt {2}\, \left (y \left (x \right )+\sqrt {4 a x +y \left (x \right )^{2}}\right ) \operatorname {arcsinh}\left (\frac {y \left (x \right )+\sqrt {4 a x +y \left (x \right )^{2}}}{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 \frac {c_{1} \left (-y \left (x \right )+\sqrt {4 a x +y \left (x \right )^{2}}\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}}}}+x -\frac {\left (-y \left (x \right )+\sqrt {4 a x +y \left (x \right )^{2}}\right ) \operatorname {arcsinh}\left (\frac {-y \left (x \right )+\sqrt {4 a x +y \left (x \right )^{2}}}{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 \end{align*}
✓ Solution by Mathematica
Time used: 1.393 (sec). Leaf size: 71
DSolve[-x - y[x]*y'[x] + a*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\left \{x=-\frac {a K[1] \log \left (\sqrt {K[1]^2+1}-K[1]\right )}{\sqrt {K[1]^2+1}}+\frac {c_1 K[1]}{\sqrt {K[1]^2+1}},y(x)=a K[1]-\frac {x}{K[1]}\right \},\{y(x),K[1]\}\right ] \]