23.4.230 problem 230

Internal problem ID [6532]
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 : 230
Date solved : Friday, October 03, 2025 at 02:09:27 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} x \left (1+x \right )^{2} y y^{\prime \prime }&=a \left (2+x \right ) y^{2}-2 \left (x^{2}+1\right ) y y^{\prime }+x \left (1+x \right )^{2} {y^{\prime }}^{2} \end{align*}

✓ Maple. Time used: 0.179 (sec). Leaf size: 80

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

\begin{align*} y &= 0 \\ y &= c_2 \,x^{a} {\mathrm e}^{\frac {16 a \int \frac {{\mathrm e}^{-\frac {4}{1+x}} \operatorname {Ei}_{1}\left (-\frac {4}{1+x}\right )}{x^{2}}d x x -16 \,{\mathrm e}^{-4} \operatorname {Ei}_{1}\left (-\frac {4 x}{1+x}\right ) c_1 x -4 c_1 \left (1+x \right ) {\mathrm e}^{-\frac {4}{1+x}}-5 a}{4 x}} \\ \end{align*}

✓ Mathematica. Time used: 60.278 (sec). Leaf size: 70

ode=x*(1 + x)^2*y[x]*D[y[x],{x,2}] == a*(2 + x)*y[x]^2 - 2*(1 + x^2)*y[x]*D[y[x],x] + x*(1 + x)^2*D[y[x],x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_2 \exp \left (\int _1^x\frac {e^{-\frac {4}{K[1]+1}} \left (4 c_1-16 a \operatorname {ExpIntegralEi}\left (\frac {4}{K[1]+1}\right )+a e^{\frac {4}{K[1]+1}} (4 K[1]+5)\right )}{4 K[1]^2}dK[1]\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a*(x + 2)*y(x)**2 + x*(x + 1)**2*y(x)*Derivative(y(x), (x, 2)) - x*(x + 1)**2*Derivative(y(x), x)**2 + (2*x**2 + 2)*y(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (x**2*y(x) + sqrt((-a*x**4*y(x) - 4*a*x**3