23.4.103 problem 103

Internal problem ID [6405]
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 : 103
Date solved : Friday, October 03, 2025 at 02:05:39 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.041 (sec). Leaf size: 65

ode:=x^2*diff(diff(y(x),x),x) = (b*y(x)^2+a*x^2*diff(y(x),x)^2)^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ y-{\mathrm e}^{\int _{}^{\ln \left (x \right )}\operatorname {RootOf}\left (y \int _{}^{\textit {\_Z}}-\frac {1}{\textit {\_a}^{2} y-\textit {\_a} y-\sqrt {y^{2} \left (\textit {\_a}^{2} a +b \right )}}d \textit {\_a} -\textit {\_b} +c_1 \right )d \textit {\_b} +c_2} &= 0 \\ \end{align*}

✗ Mathematica

ode=x^2*D[y[x],{x,2}] == Sqrt[b*y[x]^2 + a*x^2*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(x**2*Derivative(y(x), (x, 2)) - sqrt(a*x**2*Derivative(y(x), x)**2 + b*y(x)**2),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - sqrt((-b*y(x)**2 + x**4*Derivative(y(x), (