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: 0.062 (sec). Leaf size: 269
dsolve(a*diff(y(x),x)^2-y(x)*diff(y(x),x)-x = 0,y(x), singsol=all)
\begin{align*} \frac {-\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}+x \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}}}+c_{1} y \left (x \right )+\sqrt {4 a x +y \left (x \right )^{2}}\, c_{1}}{\sqrt {\frac {y \left (x \right ) \sqrt {4 a x +y \left (x \right )^{2}}+y \left (x \right )^{2}+2 a \left (a +x \right )}{a^{2}}}} &= 0 \\ \frac {-\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}-\frac {c_{1} \sqrt {2}\, y \left (x \right )}{2}+\frac {c_{1} \sqrt {2}\, \sqrt {4 a x +y \left (x \right )^{2}}}{2}+x \sqrt {\frac {y \left (x \right )^{2}-y \left (x \right ) \sqrt {4 a x +y \left (x \right )^{2}}+2 a^{2}+2 a x}{a^{2}}}}{\sqrt {\frac {-y \left (x \right ) \sqrt {4 a x +y \left (x \right )^{2}}+y \left (x \right )^{2}+2 a \left (a +x \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]
\[ \text {Solve}\left [\left \{x=-\frac {a K[1] \log \left (\sqrt {K[1]^2+1}-K[1]\right )}{\sqrt {K[1]^2+1}}+\frac {c_1 K[1]}{\sqrt {K[1]^2+1}},y(x)=a K[1]-\frac {x}{K[1]}\right \},\{y(x),K[1]\}\right ] \]