Internal problem ID [9335]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1757 (book 6.166).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {y y^{\prime \prime } a +b \left (y^{\prime }\right )^{2}-\frac {y y^{\prime }}{\sqrt {c^{2}+x^{2}}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 79
dsolve(a*y(x)*diff(diff(y(x),x),x)+b*diff(y(x),x)^2-y(x)*diff(y(x),x)/(c^2+x^2)^(1/2)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ y \relax (x ) = \left (\frac {a}{\left (a +b \right ) \left (\frac {c_{1} 2^{\frac {1}{a}} a \,x^{\frac {1}{a}+1} \hypergeom \left (\left [-\frac {1}{2 a}, -\frac {1}{2 a}-\frac {1}{2}\right ], \left [1-\frac {1}{a}\right ], -\frac {c^{2}}{x^{2}}\right )}{a +1}+c_{2}\right )}\right )^{-\frac {a}{a +b}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.657 (sec). Leaf size: 143
DSolve[-((y[x]*y'[x])/Sqrt[c^2 + x^2]) + b*y'[x]^2 + a*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2 \exp \left (\int _1^x\frac {\left (1-\frac {K[2]}{\sqrt {c^2+K[2]^2}}\right )^{\left .-\frac {1}{2}\right /a} \left (\frac {K[2]}{\sqrt {c^2+K[2]^2}}+1\right )^{\left .\frac {1}{2}\right /a}}{c_1-\int _1^{K[2]}-\frac {(a+b) \left (1-\frac {K[1]}{\sqrt {c^2+K[1]^2}}\right )^{\left .-\frac {1}{2}\right /a} \left (\frac {K[1]}{\sqrt {c^2+K[1]^2}}+1\right )^{\left .\frac {1}{2}\right /a}}{a}dK[1]}dK[2]\right ) \\ \end{align*}