Internal problem ID [9232]
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: 0.703 (sec). Leaf size: 190
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 {4^{\frac {1}{3}} \left (\left (4 c_{1} +3 x \right )^{2}\right )^{\frac {1}{3}}}{4 c_{1} +3 x} \\ y \left (x \right ) = -\frac {4^{\frac {1}{3}} \left (\left (4 c_{1} +3 x \right )^{2}\right )^{\frac {1}{3}}}{2 \left (4 c_{1} +3 x \right )}-\frac {i \sqrt {3}\, 4^{\frac {1}{3}} \left (\left (4 c_{1} +3 x \right )^{2}\right )^{\frac {1}{3}}}{2 \left (4 c_{1} +3 x \right )} \\ y \left (x \right ) = -\frac {4^{\frac {1}{3}} \left (\left (4 c_{1} +3 x \right )^{2}\right )^{\frac {1}{3}}}{2 \left (4 c_{1} +3 x \right )}+\frac {i \sqrt {3}\, 4^{\frac {1}{3}} \left (\left (4 c_{1} +3 x \right )^{2}\right )^{\frac {1}{3}}}{8 c_{1} +6 x} \\ y \left (x \right ) = \frac {\tan \left (c_{2} \left (\frac {1}{c_{1}^{2}}\right )^{\frac {3}{2}}+x \left (\frac {1}{c_{1}^{2}}\right )^{\frac {3}{2}}\right )}{c_{1}} \\ y \left (x \right ) = \frac {\tanh \left (c_{2} \left (\frac {1}{c_{1}^{2}}\right )^{\frac {3}{2}}+x \left (\frac {1}{c_{1}^{2}}\right )^{\frac {3}{2}}\right )}{c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.331 (sec). Leaf size: 33
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 ) \\ \end{align*}