23.4.279 problem 282

Internal problem ID [6581]
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 : 282
Date solved : Tuesday, September 30, 2025 at 03:11:08 PM
CAS classification : [[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_poly_yn]]

\begin{align*} y+x y^{\prime }+2 \left (x +y\right ) {y^{\prime }}^{2}+\left (y^{2}+2 x^{2} y^{\prime }\right ) y^{\prime \prime }&=0 \end{align*}

✗ Maple

ode:=y(x)+x*diff(y(x),x)+2*(x+y(x))*diff(y(x),x)^2+(y(x)^2+2*x^2*diff(y(x),x))*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=y[x] + x*D[y[x],x] + 2*(x + y[x])*D[y[x],x]^2 + (y[x]^2 + 2*x^2*D[y[x],x])*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-x*(2*x*Derivative(y(x), (x, 2)) + 1) + s