23.4.112 problem 112

Internal problem ID [6414]
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 : 112
Date solved : Friday, October 03, 2025 at 02:05:41 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} x^{4} y^{\prime \prime }&=-4 y^{2}+x^{2} y^{\prime } \left (x +y^{\prime }\right ) \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 32

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

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

✓ Mathematica. Time used: 0.439 (sec). Leaf size: 189

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

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

✗ Sympy

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

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