60.7.166 problem 1757 (book 6.166)

Internal problem ID [11755]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1757 (book 6.166)
Date solved : Monday, January 27, 2025 at 11:34:13 PM
CAS classification : [_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} a y y^{\prime \prime }+b {y^{\prime }}^{2}-\frac {y y^{\prime }}{\sqrt {c^{2}+x^{2}}}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.055 (sec). Leaf size: 82

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 &= 0 \\ y &= {\left (\frac {a \left (a +1\right )}{\left (a +b \right ) \left (c_{1} 2^{\frac {1}{a}} a \,x^{\frac {a +1}{a}} \operatorname {hypergeom}\left (\left [-\frac {1}{2 a}, -\frac {a +1}{2 a}\right ], \left [\frac {a -1}{a}\right ], -\frac {c^{2}}{x^{2}}\right )+c_{2} a +c_{2} \right )}\right )}^{-\frac {a}{a +b}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 60.771 (sec). Leaf size: 120

DSolve[-((y[x]*D[y[x],x])/Sqrt[c^2 + x^2]) + b*D[y[x],x]^2 + a*y[x]*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_2 \exp \left (\int _1^x\frac {\left (a^2-1\right ) \left (K[1]+\sqrt {c^2+K[1]^2}\right )^{1+\frac {1}{a}}}{(a+b) \left ((a-1) K[1] \left (K[1]+\sqrt {c^2+K[1]^2}\right )-c^2\right ) \left (K[1]+\sqrt {c^2+K[1]^2}\right )^{\frac {1}{a}}+\left (a^2-1\right ) c_1 \left (K[1]+\sqrt {c^2+K[1]^2}\right )}dK[1]\right ) \]