Internal problem ID [9409]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1831 (book 6.240).
ODE order: 2.
ODE degree: 2.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} \left (2-9 x \right ) \left (y^{\prime \prime }\right )^{2}-6 x \left (1-6 x \right ) y^{\prime } y^{\prime \prime }+6 y^{\prime \prime } y-36 x \left (y^{\prime }\right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.312 (sec). Leaf size: 316
dsolve(x^2*(2-9*x)*diff(diff(y(x),x),x)^2-6*x*(1-6*x)*diff(y(x),x)*diff(diff(y(x),x),x)+6*diff(diff(y(x),x),x)*y(x)-36*x*diff(y(x),x)^2=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 27 c_{1} \left (\left (9 x -1\right ) \sqrt {9}+9 \sqrt {\left (9 x -2\right ) x}\right )^{-\frac {5 \sqrt {9}}{18}} \sqrt {\frac {\frac {\left (-\frac {1}{2}+\frac {5 x}{2}\right ) \sqrt {16}}{2 \sqrt {\left (9 x -2\right ) x}}+1}{\sqrt {\frac {-16 x^{2}+8 x -1}{x \left (9 x -2\right )}}}}\, \left (\left (9 x -1\right ) \sqrt {9}+9 \sqrt {9 x^{2}-2 x}\right )^{-\frac {2 \sqrt {9}}{9}} \sqrt {4 x -1}\, x \,{\mathrm e}^{-\frac {\sqrt {16}\, \sqrt {\left (9 x -2\right ) x}}{2}+2 \sqrt {9 x^{2}-2 x}} \\ y \relax (x ) = \frac {c_{1} \left (\left (9 x -1\right ) \sqrt {9}+9 \sqrt {\left (9 x -2\right ) x}\right )^{\frac {5 \sqrt {9}}{18}} \left (\left (9 x -1\right ) \sqrt {9}+9 \sqrt {9 x^{2}-2 x}\right )^{\frac {2 \sqrt {9}}{9}} \sqrt {4 x -1}\, x \,{\mathrm e}^{\frac {\sqrt {16}\, \sqrt {\left (9 x -2\right ) x}}{2}-2 \sqrt {9 x^{2}-2 x}}}{27 \sqrt {\frac {\frac {\left (-\frac {1}{2}+\frac {5 x}{2}\right ) \sqrt {16}}{2 \sqrt {\left (9 x -2\right ) x}}+1}{\sqrt {\frac {-16 x^{2}+8 x -1}{x \left (9 x -2\right )}}}}} \\ y \relax (x ) = 0 \\ y \relax (x ) = x^{3} c_{1}+x c_{2}+\frac {c_{2}^{2}}{c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.017 (sec). Leaf size: 24
DSolve[-36*x*y'[x]^2 + 6*y[x]*y''[x] - 6*(1 - 6*x)*x*y'[x]*y''[x] + (2 - 9*x)*x^2*y''[x]^2 == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_1{}^2 x^3}{c_2}+c_1 x+c_2 \\ \end{align*}