56.38.8 problem Ex 8

Internal problem ID [14300]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IX, Miscellaneous methods for solving equations of higher order than first. Article 62. Summary. Page 144
Problem number : Ex 8
Date solved : Thursday, October 02, 2025 at 09:30:20 AM
CAS classification : [[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} x^{2}+2 y+4 \left (x +y\right ) y^{\prime }+2 {y^{\prime }}^{2} x +x \left (x +2 y\right ) y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.027 (sec). Leaf size: 80

ode:=x*(2*y(x)+x)*diff(diff(y(x),x),x)+2*x*diff(y(x),x)^2+4*(x+y(x))*diff(y(x),x)+2*y(x)+x^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {-3 x^{2}+\sqrt {3}\, \sqrt {-x \left (x^{4}-3 x^{3}+12 c_2 x -12 c_1 \right )}}{6 x} \\ y &= \frac {-3 x^{2}-\sqrt {3}\, \sqrt {-x \left (x^{4}-3 x^{3}+12 c_2 x -12 c_1 \right )}}{6 x} \\ \end{align*}

✓ Mathematica. Time used: 1.075 (sec). Leaf size: 104

ode=x*(x+2*y[x])*D[y[x],{x,2}]+2*x*(D[y[x],x])^2+4*(x+y[x])*D[y[x],x]+2*y[x]+x^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{6} \left (-3 x-\sqrt {3} \sqrt {\frac {1}{x^2}} \sqrt {x \left (-x^4+3 x^3+12 c_2 x+12 c_1\right )}\right )\\ y(x)&\to \frac {1}{6} \left (-3 x+\sqrt {3} \sqrt {\frac {1}{x^2}} \sqrt {x \left (-x^4+3 x^3+12 c_2 x+12 c_1\right )}\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2 + x*(x + 2*y(x))*Derivative(y(x), (x, 2)) + 2*x*Derivative(y(x), x)**2 + (4*x + 4*y(x))*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 + sqrt(-2*x**3*Derivative(y(x), (x, 2)) - 2*x**3 - 4*x**2*y(x)*Derivative(y(x), (x, 2)) + 4*x**2 + 4*x*y(x) + 4*y(x)**2)/2 - y(x))/x cannot be solved by the factorable group method