Internal problem ID [10153]
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]]
\[ \boxed {x^{2} \left (2-9 x \right ) {y^{\prime \prime }}^{2}-6 x \left (-6 x +1\right ) y^{\prime } y^{\prime \prime }+6 y^{\prime \prime } y-36 x {y^{\prime }}^{2}=0} \]
✓ Solution by Maple
Time used: 0.297 (sec). Leaf size: 232
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 \left (x \right ) &= \frac {9 c_{1} \sqrt {\frac {-1+5 x +\sqrt {9 x^{2}-2 x}}{\sqrt {9 x^{2}-2 x}\, \sqrt {-\frac {\left (4 x -1\right )^{2}}{x \left (9 x -2\right )}}}}\, \sqrt {4 x -1}\, x}{\left (-1+9 x +3 \sqrt {9 x^{2}-2 x}\right ) \sqrt {27 x -3+9 \sqrt {9 x^{2}-2 x}}} \\ y \left (x \right ) &= \frac {c_{1} \left (-1+9 x +3 \sqrt {9 x^{2}-2 x}\right ) \sqrt {27 x -3+9 \sqrt {9 x^{2}-2 x}}\, \sqrt {4 x -1}\, x}{9 \sqrt {\frac {-1+5 x +\sqrt {9 x^{2}-2 x}}{\sqrt {9 x^{2}-2 x}\, \sqrt {-\frac {\left (4 x -1\right )^{2}}{x \left (9 x -2\right )}}}}} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= c_{1} x^{3}+c_{2} x +\frac {c_{2}^{2}}{c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.065 (sec). Leaf size: 29
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 \\ y(x)\to \text {Indeterminate} \\ \end{align*}