54.7.13 problem 1603 (6.13)

Internal problem ID [12862]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1603 (6.13)
Date solved : Friday, October 03, 2025 at 03:47:34 AM
CAS classification : [NONE]

\begin{align*} y^{\prime \prime }-\frac {1}{\left (a y^{2}+b x y+c \,x^{2}+\alpha y+\beta x +\gamma \right )^{{3}/{2}}}&=0 \end{align*}

✓ Maple. Time used: 1.600 (sec). Leaf size: 1016

ode:=diff(diff(y(x),x),x)-1/(a*y(x)^2+b*x*y(x)+c*x^2+alpha*y(x)+beta*x+gamma)^(3/2) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Mathematica

ode=D[y[x],{x,2}]-(a*y[x]^2+b*x*y[x]+c*x^2+\[Alpha]*y[x]+\[Beta]*x+\[Gamma])^(-3/2) == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
Alpha = symbols("Alpha") 
BETA = symbols("BETA") 
Gamma = symbols("Gamma") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) - 1/(Alpha*y(x) + BETA*x + Gamma + a*y(x)**2 + b*x*y(x) + c*x**2)**(3/2),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : solve: Cannot solve Derivative(y(x), (x, 2)) - 1/(Alpha*y(x) + BETA*x + Gamma + a*y(x)**2 + b*x*y(x) + c*x**2)**(3/2)