23.4.244 problem 244

Internal problem ID [6546]
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 : 244
Date solved : Friday, October 03, 2025 at 02:09:29 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1]]

\begin{align*} \left (x +y^{2}\right ) y^{\prime \prime }&=2 \left (x -y^{2}\right ) {y^{\prime }}^{3}-y^{\prime } \left (1+4 y y^{\prime }\right ) \end{align*}

✓ Maple. Time used: 0.049 (sec). Leaf size: 37

ode:=(x+y(x)^2)*diff(diff(y(x),x),x) = 2*(x-y(x)^2)*diff(y(x),x)^3-diff(y(x),x)*(1+4*y(x)*diff(y(x),x)); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \sqrt {-x} \\ y &= -\sqrt {-x} \\ c_1 y+\ln \left (x +y^{2}\right )-c_2 +2 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.504 (sec). Leaf size: 26

ode=(x + y[x]^2)*D[y[x],{x,2}] == 2*(x - y[x]^2)*D[y[x],x]^3 - D[y[x],x]*(1 + 4*y[x]*D[y[x],x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [x=-y(x)^2+c_2 e^{e^{-c_1} y(x)},y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x + y(x)**2)*Derivative(y(x), (x, 2)) - (2*x - 2*y(x)**2)*Derivative(y(x), x)**3 + (4*y(x)*Derivative(y(x), x) + 1)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out