Internal problem ID [9980]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1658 (book 6.67).
ODE order: 2.
ODE degree: 2.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
\[ \boxed {y^{\prime \prime }+y^{3} y^{\prime }-y y^{\prime } \sqrt {y^{4}+4 y^{\prime }}=0} \]
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 133
dsolve(diff(diff(y(x),x),x)+y(x)^3*diff(y(x),x)-y(x)*diff(y(x),x)*(y(x)^4+4*diff(y(x),x))^(1/2)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {2^{\frac {2}{3}} \left (\left (4 c_{1} +3 x \right )^{2}\right )^{\frac {1}{3}}}{4 c_{1} +3 x} \\ y \left (x \right ) &= -\frac {2^{\frac {2}{3}} \left (\left (4 c_{1} +3 x \right )^{2}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{8 c_{1} +6 x} \\ y \left (x \right ) &= \frac {2^{\frac {2}{3}} \left (\left (4 c_{1} +3 x \right )^{2}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{8 c_{1} +6 x} \\ y \left (x \right ) &= \tan \left (\frac {c_{2} +x}{c_{1}^{3}}\right ) \sqrt {\frac {1}{c_{1}^{2}}} \\ y \left (x \right ) &= \tanh \left (\frac {c_{2} +x}{c_{1}^{3}}\right ) \sqrt {\frac {1}{c_{1}^{2}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.613 (sec). Leaf size: 38
DSolve[y[x]^3*y'[x] - y[x]*y'[x]*Sqrt[y[x]^4 + 4*y'[x]] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt {2} e^{c_1} \tan \left (2 \sqrt {2} e^{3 c_1} (x+c_2)\right ) \\ y(x)\to 0 \\ \end{align*}