54.3.112 problem 1126

Internal problem ID [12407]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1126
Date solved : Wednesday, October 01, 2025 at 01:44:32 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+\left (2 a \,x^{3}-1\right ) y^{\prime }+\left (a^{2} x^{3}+a \right ) x^{2} y&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 19

ode:=x*diff(diff(y(x),x),x)+(2*a*x^3-1)*diff(y(x),x)+(a^2*x^3+a)*x^2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {a \,x^{3}}{3}} \left (c_2 \,x^{2}+c_1 \right ) \]

✓ Mathematica. Time used: 0.045 (sec). Leaf size: 34

ode=x*D[y[x],{x,2}]+(2*a*x^3-1)*D[y[x],x]+(a^2*x^3+a)*x^2*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

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

False