Internal problem ID [9402]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1824 (book 6.233).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (\left (y^{\prime }\right )^{2}+a \left (y^{\prime } x -y\right )\right ) y^{\prime \prime }-b=0} \end {gather*}
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 423
dsolve((diff(y(x),x)^2+a*(x*diff(y(x),x)-y(x)))*diff(diff(y(x),x),x)-b=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {a \,x^{2}}{4}+\RootOf \left (-x +\int _{}^{\textit {\_Z}}\frac {\sqrt {\textit {\_f}^{3} a^{3}-4 \textit {\_f}^{2} a b +2 \textit {\_f} a c_{1}+\sqrt {4 \textit {\_f} b -2 c_{1}}\, \textit {\_f}^{2} a^{2}-4 \sqrt {4 \textit {\_f} b -2 c_{1}}\, b \textit {\_f} +2 \sqrt {4 \textit {\_f} b -2 c_{1}}\, c_{1}}}{\textit {\_f}^{2} a^{2}-4 \textit {\_f} b +2 c_{1}}d \textit {\_f} +c_{2}\right ) \\ y \relax (x ) = -\frac {a \,x^{2}}{4}+\RootOf \left (-x +\int _{}^{\textit {\_Z}}\frac {\sqrt {\textit {\_f}^{3} a^{3}-4 \textit {\_f}^{2} a b +2 \textit {\_f} a c_{1}-\sqrt {4 \textit {\_f} b -2 c_{1}}\, \textit {\_f}^{2} a^{2}+4 \sqrt {4 \textit {\_f} b -2 c_{1}}\, b \textit {\_f} -2 \sqrt {4 \textit {\_f} b -2 c_{1}}\, c_{1}}}{\textit {\_f}^{2} a^{2}-4 \textit {\_f} b +2 c_{1}}d \textit {\_f} +c_{2}\right ) \\ y \relax (x ) = -\frac {a \,x^{2}}{4}+\RootOf \left (-x -\left (\int _{}^{\textit {\_Z}}\frac {\sqrt {\textit {\_f}^{3} a^{3}-4 \textit {\_f}^{2} a b +2 \textit {\_f} a c_{1}+\sqrt {4 \textit {\_f} b -2 c_{1}}\, \textit {\_f}^{2} a^{2}-4 \sqrt {4 \textit {\_f} b -2 c_{1}}\, b \textit {\_f} +2 \sqrt {4 \textit {\_f} b -2 c_{1}}\, c_{1}}}{\textit {\_f}^{2} a^{2}-4 \textit {\_f} b +2 c_{1}}d \textit {\_f} \right )+c_{2}\right ) \\ y \relax (x ) = -\frac {a \,x^{2}}{4}+\RootOf \left (-x -\left (\int _{}^{\textit {\_Z}}\frac {\sqrt {\textit {\_f}^{3} a^{3}-4 \textit {\_f}^{2} a b +2 \textit {\_f} a c_{1}-\sqrt {4 \textit {\_f} b -2 c_{1}}\, \textit {\_f}^{2} a^{2}+4 \sqrt {4 \textit {\_f} b -2 c_{1}}\, b \textit {\_f} -2 \sqrt {4 \textit {\_f} b -2 c_{1}}\, c_{1}}}{\textit {\_f}^{2} a^{2}-4 \textit {\_f} b +2 c_{1}}d \textit {\_f} \right )+c_{2}\right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.181 (sec). Leaf size: 281
DSolve[-b + (y'[x]^2 + a*(-y[x] + x*y'[x]))*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [-\int \frac {a \left (\frac {a x^2}{4}+y(x)\right )+\sqrt {4 b \left (\frac {a x^2}{4}+y(x)\right )-2 c_1}}{\sqrt {\left (a^2 \left (\frac {a x^2}{4}+y(x)\right )^2-4 b \left (\frac {a x^2}{4}+y(x)\right )+2 c_1\right ) \left (a \left (\frac {a x^2}{4}+y(x)\right )+\sqrt {4 b \left (\frac {a x^2}{4}+y(x)\right )-2 c_1}\right )}}d\left (\frac {a x^2}{4}+y(x)\right )=-x+c_2,y(x)\right ] \\ \text {Solve}\left [\int \frac {a \left (\frac {a x^2}{4}+y(x)\right )+\sqrt {4 b \left (\frac {a x^2}{4}+y(x)\right )-2 c_1}}{\sqrt {\left (a^2 \left (\frac {a x^2}{4}+y(x)\right )^2-4 b \left (\frac {a x^2}{4}+y(x)\right )+2 c_1\right ) \left (a \left (\frac {a x^2}{4}+y(x)\right )+\sqrt {4 b \left (\frac {a x^2}{4}+y(x)\right )-2 c_1}\right )}}d\left (\frac {a x^2}{4}+y(x)\right )=-x+c_2,y(x)\right ] \\ \end{align*}