Internal problem ID [7985]
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]
Solve \begin {gather*} \boxed {a \left (y^{\prime }\right )^{2}+y y^{\prime }-x=0} \end {gather*}
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 376
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 {-y \relax (x )+\sqrt {4 x a +y \relax (x )^{2}}+2 a}{a}}\, \sqrt {\frac {-y \relax (x )+\sqrt {4 x a +y \relax (x )^{2}}-2 a}{a}}}+x +\frac {\left (-y \relax (x )+\sqrt {4 x a +y \relax (x )^{2}}\right ) \ln \left (\frac {\sqrt {\frac {4 x a +2 y \relax (x )^{2}-2 y \relax (x ) \sqrt {4 x a +y \relax (x )^{2}}-4 a^{2}}{a^{2}}}\, a +\sqrt {4 x a +y \relax (x )^{2}}-y \relax (x )}{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 {-2 y \relax (x )-2 \sqrt {4 x a +y \relax (x )^{2}}+4 a}{a}}\, \sqrt {\frac {-2 y \relax (x )-2 \sqrt {4 x a +y \relax (x )^{2}}-4 a}{a}}}+x -\frac {\left (y \relax (x )+\sqrt {4 x a +y \relax (x )^{2}}\right ) \sqrt {2}\, \ln \left (-\frac {-\sqrt {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}}}\, a +\sqrt {4 x a +y \relax (x )^{2}}+y \relax (x )}{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: 0.802 (sec). Leaf size: 61
DSolve[-x + y[x]*y'[x] + a*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\left \{x=\frac {a K[1] \text {ArcSin}(K[1])}{\sqrt {1-K[1]^2}}+\frac {c_1 K[1]}{\sqrt {1-K[1]^2}},y(x)=\frac {x}{K[1]}-a K[1]\right \},\{y(x),K[1]\}\right ] \]