23.4.87 problem 87

Internal problem ID [6389]
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 : 87
Date solved : Friday, October 03, 2025 at 02:05:36 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

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

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

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

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

✓ Mathematica. Time used: 0.289 (sec). Leaf size: 160

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

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

✗ Sympy

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

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