23.4.283 problem 286

Internal problem ID [6585]
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 : 286
Date solved : Tuesday, September 30, 2025 at 03:13:59 PM
CAS classification : [[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_poly_yn]]

\begin{align*} y+3 x y^{\prime }+2 y {y^{\prime }}^{3}+\left (x^{2}+2 y^{2} y^{\prime }\right ) y^{\prime \prime }&=0 \end{align*}

✗ Maple

ode:=y(x)+3*x*diff(y(x),x)+2*y(x)*diff(y(x),x)^3+(x^2+2*y(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] + 3*x*D[y[x],x] + 2*y[x]*D[y[x],x]^3 + (x^2 + 2*y[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(3*x*Derivative(y(x), x) + (x**2 + 2*y(x)**2*Derivative(y(x), x))*Derivative(y(x), (x, 2)) + 2*y(x)*Derivative(y(x), x)**3 + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -(3*x + 2*y(x)**2*Derivative(y(x), (x, 2)))/(2*(-1/2 + sqrt(3)*I