60.7.240 problem 1831 (book 6.240)

Internal problem ID [11829]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1831 (book 6.240)
Date solved : Tuesday, January 28, 2025 at 06:23:52 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (2-9 x \right ) {y^{\prime \prime }}^{2}-6 x \left (1-6 x \right ) y^{\prime } y^{\prime \prime }+6 y^{\prime \prime } y-36 x {y^{\prime }}^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 1.022 (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 &= \frac {9 c_{1} \sqrt {\frac {5 x -1+\sqrt {9 x^{2}-2 x}}{\sqrt {9 x^{2}-2 x}\, \sqrt {-\frac {\left (-1+4 x \right )^{2}}{x \left (9 x -2\right )}}}}\, \sqrt {-1+4 x}\, 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 &= \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 {-1+4 x}\, x}{9 \sqrt {\frac {5 x -1+\sqrt {9 x^{2}-2 x}}{\sqrt {9 x^{2}-2 x}\, \sqrt {-\frac {\left (-1+4 x \right )^{2}}{x \left (9 x -2\right )}}}}} \\ y &= 0 \\ y &= c_{1} x^{3}+c_{2} x +\frac {c_{2}^{2}}{c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.052 (sec). Leaf size: 29

DSolve[-36*x*D[y[x],x]^2 + 6*y[x]*D[y[x],{x,2}] - 6*(1 - 6*x)*x*D[y[x],x]*D[y[x],{x,2}] + (2 - 9*x)*x^2*D[y[x],{x,2}]^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*}