60.7.232 problem 1823 (book 6.232)

Internal problem ID [11821]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1823 (book 6.232)
Date solved : Monday, January 27, 2025 at 11:40:19 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} \left ({y^{\prime }}^{2}+y^{2}\right ) y^{\prime \prime }+y^{3}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.115 (sec). Leaf size: 163

dsolve((diff(y(x),x)^2+y(x)^2)*diff(diff(y(x),x),x)+y(x)^3=0,y(x), singsol=all)
 

\begin{align*} y &= 0 \\ y &= \frac {\sqrt {c_{1} +\tan \left (\sqrt {3}\, x \right )}\, {\mathrm e}^{-\frac {\sqrt {3}\, \left (\int \frac {\sqrt {\left (9 c_{1}^{2}+12\right ) \sec \left (\sqrt {3}\, x \right )^{2}+3 c_{1}^{2}+6 c_{1} \tan \left (\sqrt {3}\, x \right )-3}}{c_{1} +\tan \left (\sqrt {3}\, x \right )}d x \right )}{6}+c_{2}}}{\left (\sec \left (\sqrt {3}\, x \right )^{2}\right )^{{1}/{4}}} \\ y &= \frac {\sqrt {c_{1} +\tan \left (\sqrt {3}\, x \right )}\, {\mathrm e}^{\frac {\sqrt {3}\, \left (\int \frac {\sqrt {\left (9 c_{1}^{2}+12\right ) \sec \left (\sqrt {3}\, x \right )^{2}+3 c_{1}^{2}+6 c_{1} \tan \left (\sqrt {3}\, x \right )-3}}{c_{1} +\tan \left (\sqrt {3}\, x \right )}d x \right )}{6}+c_{2}}}{\left (\sec \left (\sqrt {3}\, x \right )^{2}\right )^{{1}/{4}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.678 (sec). Leaf size: 63

DSolve[y[x]^3 + (y[x]^2 + D[y[x],x]^2)*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_2 \exp \left (\int _1^x\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {K[1]^2+1}{\left (K[1]^2-K[1]+1\right ) \left (K[1]^2+K[1]+1\right )}dK[1]\&\right ][c_1-K[2]]dK[2]\right ) \]