60.7.68 problem 1677 (book 6.86)

Internal problem ID [11618]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1677 (book 6.86)
Date solved : Thursday, March 13, 2025 at 09:21:33 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.085 (sec). Leaf size: 75

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

\[ y = \frac {\left (-\sqrt {-a b}\, \left (\int \frac {\operatorname {BesselY}\left (1, \sqrt {-a b}\, x \right ) c_{1} +\operatorname {BesselJ}\left (1, \sqrt {-a b}\, x \right )}{x \left (c_{1} \operatorname {BesselY}\left (0, \sqrt {-a b}\, x \right )+\operatorname {BesselJ}\left (0, \sqrt {-a b}\, x \right )\right )}d x \right )+c_{2} a \right ) x}{a} \]

✓ Mathematica. Time used: 120.231 (sec). Leaf size: 118

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

\[ y(x)\to x \left (\int _1^x\frac {i \sqrt {b} \left (\operatorname {BesselY}\left (1,-i \sqrt {a} \sqrt {b} K[1]\right )-\operatorname {BesselJ}\left (1,i \sqrt {a} \sqrt {b} K[1]\right ) c_1\right )}{\sqrt {a} \left (\operatorname {BesselY}\left (0,-i \sqrt {a} \sqrt {b} K[1]\right )+\operatorname {BesselJ}\left (0,i \sqrt {a} \sqrt {b} K[1]\right ) c_1\right ) K[1]}dK[1]+c_2\right ) \]

✗ 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**2 + x**2*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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