83.34.4 problem 4

Internal problem ID [19346]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VII. Exact differential equations and certain particular forms of equations. Exercise VII (H) at page 118
Problem number : 4
Date solved : Monday, March 31, 2025 at 07:09:09 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }&=\sqrt {m \,x^{2} {y^{\prime }}^{3}+n y^{2}} \end{align*}

✗ Maple

ode:=x^2*diff(diff(y(x),x),x) = (m*x^2*diff(y(x),x)^3+n*y(x)^2)^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

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

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
m = symbols("m") 
n = symbols("n") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - sqrt(m*x**2*Derivative(y(x), x)**3 + n*y(x)**2),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE (-n*y(x)**2/(m*x**2) + x**2*Derivative(y(x), (x, 2))**2/m)**(1/3)/2 - sqrt(3)*I*(-n*y(x)**2/(m*x**2) + x**2*Derivative(y(x), (x, 2))**2/m)**(1/3)/2 + Derivative(y(x), x) cannot be solved by the factorable group method