82.48.1 problem Ex. 1

Internal problem ID [18911]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VIII. End of chapter problems at page 107
Problem number : Ex. 1
Date solved : Monday, March 31, 2025 at 06:21:43 PM
CAS classification : [[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_poly_yn]]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

ode=(y[x]^2+2*x^2*D[y[x],x]*D[y[x],{x,2}])+2*(y[x]+x)*D[y[x],x]^2+x*D[y[x],x]+y[x]==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) + sqrt(4*x**4*Derivative(y(x), (x, 2))**2 + 4*x**3*Derivative(y(x), (x, 2)) + x**2 - 8*x*y(x)**2*Derivative(y(x), (x, 2)) - 8*x*y(x) - 8*y(x)**3*Derivative(y(x), (x, 2)) - 8*y(x)**2))/(4*(x + y(x))) cannot be solved by the factorable group method