54.3.304 problem 1321

Internal problem ID [12599]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1321
Date solved : Wednesday, October 01, 2025 at 02:12:47 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x +1\right ) y^{\prime \prime }-x \left (2 x +1\right ) y^{\prime }+\left (2 x +1\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 15

ode:=x^2*(1+x)*diff(diff(y(x),x),x)-x*(2*x+1)*diff(y(x),x)+(2*x+1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = x \left (c_2 \ln \left (x \right )+c_2 x +c_1 \right ) \]

✓ Mathematica. Time used: 0.104 (sec). Leaf size: 95

ode=(1 + 2*x)*y[x] - x*(1 + 2*x)*D[y[x],x] + x^2*(1 + x)*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \exp \left (\int _1^x\frac {1}{2 K[1]^2+2 K[1]}dK[1]-\frac {1}{2} \int _1^x-\frac {2 K[2]+1}{K[2]^2+K[2]}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {1}{2 K[1]^2+2 K[1]}dK[1]\right )dK[3]+c_1\right ) \end{align*}

✗ Sympy

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

False