23.4.207 problem 207

Internal problem ID [6509]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 207
Date solved : Tuesday, September 30, 2025 at 03:02:30 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y y^{\prime \prime }&=b^{2} x y^{3}+a y y^{\prime }+x {y^{\prime }}^{2} \end{align*}

✗ Maple

ode:=x*y(x)*diff(diff(y(x),x),x) = b^2*x*y(x)^3+a*y(x)*diff(y(x),x)+x*diff(y(x),x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=x*y[x]*D[y[x],{x,2}] == b^2*x*y[x]^3 + a*y[x]*D[y[x],x] + x*D[y[x],x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(-a*y(x)*Derivative(y(x), x) - b**2*x*y(x)**3 + x*y(x)*Derivative(y(x), (x, 2)) - 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**2*y(x) - 4*b**2*x**2*y