54.7.158 problem 1775 (book 6.184)

Internal problem ID [13007]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1775 (book 6.184)
Date solved : Friday, October 03, 2025 at 03:58:17 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} a \,x^{2} y y^{\prime \prime }+b \,x^{2} {y^{\prime }}^{2}+c x y y^{\prime }+d y^{2}&=0 \end{align*}

✓ Maple. Time used: 0.042 (sec). Leaf size: 110

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

\begin{align*} y &= 0 \\ y &= \left (\frac {x^{\frac {\sqrt {\left (-4 a -4 b \right ) d +\left (a -c \right )^{2}}-a +c}{a}} \left (a^{2}+\left (-2 c -4 d \right ) a -4 d b +c^{2}\right )}{\left (a +b \right )^{2} \left (c_1 \,x^{\frac {\sqrt {\left (-4 a -4 b \right ) d +\left (a -c \right )^{2}}}{a}}-c_2 \right )^{2}}\right )^{-\frac {a}{2 a +2 b}} \\ \end{align*}

✓ Mathematica. Time used: 29.078 (sec). Leaf size: 92

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

\begin{align*} y(x)&\to c_2 \exp \left (-\frac {\log (x) \left (a \left (\sqrt {\frac {a^2-2 a (c+2 d)-4 b d+c^2}{a^2}}-1\right )+c\right )-2 a \log \left (x^{\sqrt {\frac {a^2-2 a (c+2 d)-4 b d+c^2}{a^2}}}+c_1\right )}{2 (a+b)}\right ) \end{align*}

✗ Sympy

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

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