60.7.141 problem 1757 (book 6.166)

Internal problem ID [11691]
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 : Sunday, March 30, 2025 at 08:42:46 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*}

✓ Maple. Time used: 0.063 (sec). Leaf size: 82

ode:=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; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ y &= {\left (\frac {a \left (1+a \right )}{\left (b +a \right ) \left (c_1 2^{\frac {1}{a}} a \,x^{\frac {1+a}{a}} \operatorname {hypergeom}\left (\left [-\frac {1}{2 a}, -\frac {1+a}{2 a}\right ], \left [\frac {a -1}{a}\right ], -\frac {c^{2}}{x^{2}}\right )+c_2 a +c_2 \right )}\right )}^{-\frac {a}{b +a}} \\ \end{align*}

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

ode=-((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; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq(a*y(x)*Derivative(y(x), (x, 2)) + b*Derivative(y(x), x)**2 - y(x)*Derivative(y(x), x)/sqrt(c**2 + x**2),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (sqrt((-4*a*b*c**2*Derivative(y(x), (x, 2)) - 4*a*b*x**2*Derivative(y(x), (x, 2)) + y(x))*y(x)) + y(x))/(2*b*sqrt(c**2 + x**2)) cannot be solved by the factorable group method