60.7.63 problem 1671 (book 6.80)

Internal problem ID [11613]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1671 (book 6.80)
Date solved : Sunday, March 30, 2025 at 08:31:34 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+a \left (x y^{\prime }-y\right )^{2}-b&=0 \end{align*}

✓ Maple. Time used: 0.121 (sec). Leaf size: 38

ode:=x*diff(diff(y(x),x),x)+a*(-y(x)+x*diff(y(x),x))^2-b = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {x \left (i \sqrt {b}\, \int \frac {\tan \left (-i \sqrt {a}\, \sqrt {b}\, x +c_1 \right )}{x^{2}}d x +c_2 \sqrt {a}\right )}{\sqrt {a}} \]

✓ Mathematica. Time used: 120.332 (sec). Leaf size: 50

ode=-b + a*(-y[x] + x*D[y[x],x])^2 + x*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

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

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