Internal problem ID [9235]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1657 (book 6.66).
ODE order: 2.
ODE degree: 2.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-2 a \left (c +b x +y\right ) \left (1+\left (y^{\prime }\right )^{2}\right )^{\frac {3}{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 785
dsolve(diff(diff(y(x),x),x)=2*a*(c+b*x+y(x))*(diff(y(x),x)^2+1)^(3/2),y(x), singsol=all)
\begin{align*} y \relax (x ) = -i x +c_{1} \\ y \relax (x ) = i x +c_{1} \\ y \relax (x ) = -b x +\RootOf \left (-x +\int _{}^{\textit {\_Z}}\frac {4 a^{2} b^{2} c^{2} \textit {\_f}^{2}+4 a^{2} b^{2} c \,\textit {\_f}^{3}+a^{2} b^{2} \textit {\_f}^{4}-8 c_{1} a^{2} b^{2} c \textit {\_f} -4 c_{1} a^{2} b^{2} \textit {\_f}^{2}+4 c_{1}^{2} a^{2} b^{2}-2 \sqrt {-b^{2} \left (a^{2} \textit {\_f}^{4}+4 a^{2} c \,\textit {\_f}^{3}+4 a^{2} c^{2} \textit {\_f}^{2}-4 \textit {\_f}^{2} a^{2} c_{1}-8 \textit {\_f} \,a^{2} c c_{1}+4 a^{2} c_{1}^{2}-b^{2}-1\right )}\, a c \textit {\_f} -\sqrt {-b^{2} \left (a^{2} \textit {\_f}^{4}+4 a^{2} c \,\textit {\_f}^{3}+4 a^{2} c^{2} \textit {\_f}^{2}-4 \textit {\_f}^{2} a^{2} c_{1}-8 \textit {\_f} \,a^{2} c c_{1}+4 a^{2} c_{1}^{2}-b^{2}-1\right )}\, a \,\textit {\_f}^{2}-b^{4}+2 \sqrt {-b^{2} \left (a^{2} \textit {\_f}^{4}+4 a^{2} c \,\textit {\_f}^{3}+4 a^{2} c^{2} \textit {\_f}^{2}-4 \textit {\_f}^{2} a^{2} c_{1}-8 \textit {\_f} \,a^{2} c c_{1}+4 a^{2} c_{1}^{2}-b^{2}-1\right )}\, c_{1} a -b^{2}}{\left (a^{2} \textit {\_f}^{4}+4 a^{2} c \,\textit {\_f}^{3}+4 a^{2} c^{2} \textit {\_f}^{2}-4 \textit {\_f}^{2} a^{2} c_{1}-8 \textit {\_f} \,a^{2} c c_{1}+4 a^{2} c_{1}^{2}-b^{2}-1\right ) \left (b^{2}+1\right ) b}d \textit {\_f} +c_{2}\right ) \\ y \relax (x ) = -b x +\RootOf \left (-x +\int _{}^{\textit {\_Z}}-\frac {-4 a^{2} b^{2} c^{2} \textit {\_f}^{2}-4 a^{2} b^{2} c \,\textit {\_f}^{3}-a^{2} b^{2} \textit {\_f}^{4}+8 c_{1} a^{2} b^{2} c \textit {\_f} +4 c_{1} a^{2} b^{2} \textit {\_f}^{2}-4 c_{1}^{2} a^{2} b^{2}-2 \sqrt {-b^{2} \left (a^{2} \textit {\_f}^{4}+4 a^{2} c \,\textit {\_f}^{3}+4 a^{2} c^{2} \textit {\_f}^{2}-4 \textit {\_f}^{2} a^{2} c_{1}-8 \textit {\_f} \,a^{2} c c_{1}+4 a^{2} c_{1}^{2}-b^{2}-1\right )}\, a c \textit {\_f} -\sqrt {-b^{2} \left (a^{2} \textit {\_f}^{4}+4 a^{2} c \,\textit {\_f}^{3}+4 a^{2} c^{2} \textit {\_f}^{2}-4 \textit {\_f}^{2} a^{2} c_{1}-8 \textit {\_f} \,a^{2} c c_{1}+4 a^{2} c_{1}^{2}-b^{2}-1\right )}\, a \,\textit {\_f}^{2}+b^{4}+2 \sqrt {-b^{2} \left (a^{2} \textit {\_f}^{4}+4 a^{2} c \,\textit {\_f}^{3}+4 a^{2} c^{2} \textit {\_f}^{2}-4 \textit {\_f}^{2} a^{2} c_{1}-8 \textit {\_f} \,a^{2} c c_{1}+4 a^{2} c_{1}^{2}-b^{2}-1\right )}\, c_{1} a +b^{2}}{\left (a^{2} \textit {\_f}^{4}+4 a^{2} c \,\textit {\_f}^{3}+4 a^{2} c^{2} \textit {\_f}^{2}-4 \textit {\_f}^{2} a^{2} c_{1}-8 \textit {\_f} \,a^{2} c c_{1}+4 a^{2} c_{1}^{2}-b^{2}-1\right ) \left (b^{2}+1\right ) b}d \textit {\_f} +c_{2}\right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 26.892 (sec). Leaf size: 9706
DSolve[-(2*a*(c + b*x + y[x])*(1 + y'[x]^2)^(3/2)) + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
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