23.4.99 problem 99

Internal problem ID [6401]
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 : 99
Date solved : Friday, October 03, 2025 at 02:05:38 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} x^{2} y^{\prime \prime }&=6 y-4 x^{2} y^{2}+x^{4} {y^{\prime }}^{2} \end{align*}

✓ Maple. Time used: 0.022 (sec). Leaf size: 32

ode:=x^2*diff(diff(y(x),x),x) = 6*y(x)-4*x^2*y(x)^2+x^4*diff(y(x),x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\operatorname {RootOf}\left (-\ln \left (x \right )+c_2 -\int _{}^{\textit {\_Z}}\frac {1}{-4 \textit {\_f} +1+{\mathrm e}^{\textit {\_f}} c_1}d \textit {\_f} \right )}{x^{2}} \]

✓ Mathematica. Time used: 0.322 (sec). Leaf size: 194

ode=x^2*D[y[x],{x,2}] == 6*y[x] - 4*x^2*y[x]^2 + x^4*D[y[x],x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{y(x)}-\frac {x^2}{4 K[2] x^2+e^{x^2 K[2]} c_1-1}dK[2]-\int _1^x\left (\int _1^{y(x)}\left (\frac {K[3]^2 \left (2 e^{K[2] K[3]^2} c_1 K[2] K[3]+8 K[2] K[3]\right )}{\left (4 K[2] K[3]^2+e^{K[2] K[3]^2} c_1-1\right ){}^2}-\frac {2 K[3]}{4 K[2] K[3]^2+e^{K[2] K[3]^2} c_1-1}\right )dK[2]-\frac {2 y(x) K[3]^2+e^{K[3]^2 y(x)} c_1-1}{K[3] \left (4 y(x) K[3]^2+e^{K[3]^2 y(x)} c_1-1\right )}\right )dK[3]=c_2,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**4*Derivative(y(x), x)**2 + 4*x**2*y(x)**2 + x**2*Derivative(y(x), (x, 2)) - 6*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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