56.32.8 problem Ex 8

Internal problem ID [14266]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter VIII, Linear differential equations of the second order. Article 55. Summary. Page 129
Problem number : Ex 8
Date solved : Thursday, October 02, 2025 at 09:27:26 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 19

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

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

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 28

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

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

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-x**3/2 + x**2*Derivative(y(x), (x, 2))/2 + x*y(x) + y(x))/(x*(x + 1)) cannot be solved by the factorable group method