Internal problem ID [8741]
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]
\[ \boxed {a {y^{\prime }}^{2}+y^{\prime } y=x} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 396
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 ) \left (3 \ln \left (2\right )-2 \ln \left (\frac {2 \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}}}\, a -\left (y \left (x \right )-\sqrt {4 a x +y \left (x \right )^{2}}\right ) \sqrt {2}}{a}\right )\right ) \sqrt {2}}{4 \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}}}} &= 0 \\ \frac {c_{1} \left (y \left (x \right )+\sqrt {4 a x +y \left (x \right )^{2}}\right )}{2 \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 {\sqrt {2}\, \left (y \left (x \right )+\sqrt {4 a x +y \left (x \right )^{2}}\right ) \left (-\frac {3 \ln \left (2\right )}{2}+\ln \left (\frac {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 -\left (y \left (x \right )+\sqrt {4 a x +y \left (x \right )^{2}}\right ) \sqrt {2}}{a}\right )\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: 0.581 (sec). Leaf size: 79
DSolve[-x + y[x]*y'[x] + a*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\left \{x=-\frac {2 a K[1] \arctan \left (\frac {\sqrt {1-K[1]^2}}{K[1]+1}\right )}{\sqrt {1-K[1]^2}}+\frac {c_1 K[1]}{\sqrt {1-K[1]^2}},y(x)=\frac {x}{K[1]}-a K[1]\right \},\{y(x),K[1]\}\right ] \]