Internal problem ID [9401]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1822.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
Solve \begin {gather*} \boxed {\left (\left (y^{\prime }\right )^{2}+y^{2}\right ) y^{\prime \prime }+y^{3}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.158 (sec). Leaf size: 241
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 \relax (x ) = 0 \\ y \relax (x ) = \frac {2^{\frac {3}{4}} \sqrt {\frac {\cos \left (\sqrt {3}\, x \right ) c_{1}+\sin \left (\sqrt {3}\, x \right )}{\cos \left (\sqrt {3}\, x \right )}}\, {\mathrm e}^{-\frac {\left (\int \frac {\cos \left (\sqrt {3}\, x \right ) \sqrt {\frac {2 c_{1} \sin \left (\sqrt {3}\, x \right ) \cos \left (\sqrt {3}\, x \right )+\left (c_{1}^{2}-1\right ) \left (\cos ^{2}\left (\sqrt {3}\, x \right )\right )+3 c_{1}^{2}+4}{\cos \left (\sqrt {3}\, x \right )^{2}}}}{\cos \left (\sqrt {3}\, x \right ) c_{1}+\sin \left (\sqrt {3}\, x \right )}d x \right )}{2}+c_{2}}}{2 \left (\frac {1}{\cos \left (2 \sqrt {3}\, x \right )+1}\right )^{\frac {1}{4}}} \\ y \relax (x ) = \frac {2^{\frac {3}{4}} \sqrt {\frac {\cos \left (\sqrt {3}\, x \right ) c_{1}+\sin \left (\sqrt {3}\, x \right )}{\cos \left (\sqrt {3}\, x \right )}}\, {\mathrm e}^{\frac {\left (\int \frac {\cos \left (\sqrt {3}\, x \right ) \sqrt {\frac {2 c_{1} \sin \left (\sqrt {3}\, x \right ) \cos \left (\sqrt {3}\, x \right )+\left (c_{1}^{2}-1\right ) \left (\cos ^{2}\left (\sqrt {3}\, x \right )\right )+3 c_{1}^{2}+4}{\cos \left (\sqrt {3}\, x \right )^{2}}}}{\cos \left (\sqrt {3}\, x \right ) c_{1}+\sin \left (\sqrt {3}\, x \right )}d x \right )}{2}+c_{2}}}{2 \left (\frac {1}{\cos \left (2 \sqrt {3}\, x \right )+1}\right )^{\frac {1}{4}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.561 (sec). Leaf size: 369
DSolve[y[x]^3 + (y[x]^2 + y'[x]^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_2 \exp \left (-\frac {\text {ArcTan}\left (\frac {1+2 \text {InverseFunction}\left [\frac {\left (\sqrt {3}-i\right ) \text {ArcTan}\left (\frac {\text {$\#$1}}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right )}{\sqrt {6 \left (1-i \sqrt {3}\right )}}+\frac {\left (\sqrt {3}+i\right ) \text {ArcTan}\left (\frac {\text {$\#$1}}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right )}{\sqrt {6 \left (1+i \sqrt {3}\right )}}\&\right ][-x+c_1]{}^2}{\sqrt {3}}\right )}{2 \sqrt {3}}\right )}{\sqrt [4]{\text {InverseFunction}\left [\frac {\left (\sqrt {3}-i\right ) \text {ArcTan}\left (\frac {\text {$\#$1}}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right )}{\sqrt {6 \left (1-i \sqrt {3}\right )}}+\frac {\left (\sqrt {3}+i\right ) \text {ArcTan}\left (\frac {\text {$\#$1}}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right )}{\sqrt {6 \left (1+i \sqrt {3}\right )}}\&\right ][-x+c_1]{}^4+\text {InverseFunction}\left [\frac {\left (\sqrt {3}-i\right ) \text {ArcTan}\left (\frac {\text {$\#$1}}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right )}{\sqrt {6 \left (1-i \sqrt {3}\right )}}+\frac {\left (\sqrt {3}+i\right ) \text {ArcTan}\left (\frac {\text {$\#$1}}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right )}{\sqrt {6 \left (1+i \sqrt {3}\right )}}\&\right ][-x+c_1]{}^2+1}} \\ \end{align*}