Internal problem ID [9426]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 7, non-linear third and higher order
Problem number: 1847.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2], [_3rd_order, _reducible, _mu_poly_yn]]
Solve \begin {gather*} \boxed {\left (\left (y^{\prime }\right )^{2}+1\right ) y^{\prime \prime \prime }-3 y^{\prime } \left (y^{\prime \prime }\right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.151 (sec). Leaf size: 67
dsolve((diff(y(x),x)^2+1)*diff(diff(diff(y(x),x),x),x)-3*diff(y(x),x)*diff(diff(y(x),x),x)^2=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -i x +c_{1} \\ y \relax (x ) = i x +c_{1} \\ y \relax (x ) = -\sqrt {-c_{2}^{2}-2 x c_{2}-x^{2}+c_{1}}+c_{3} \\ y \relax (x ) = \sqrt {-c_{2}^{2}-2 x c_{2}-x^{2}+c_{1}}+c_{3} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.157 (sec). Leaf size: 65
DSolve[-3*y'[x]*y''[x]^2 + (1 + y'[x]^2)*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_3-\frac {i \sqrt {-1+c_1{}^2 (x+c_2){}^2}}{c_1} \\ y(x)\to \frac {i \sqrt {-1+c_1{}^2 (x+c_2){}^2}}{c_1}+c_3 \\ \end{align*}