54.3.144 problem 1158

Internal problem ID [12439]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1158
Date solved : Wednesday, October 01, 2025 at 01:45:01 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.159 (sec). Leaf size: 178

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

\[ y = \sqrt {x}\, \left ({\mathrm e}^{b x} \operatorname {HeunD}\left (-4 \sqrt {2}\, \sqrt {a b}, -1-4 \sqrt {2}\, \sqrt {a b}, 8 \sqrt {2}\, \sqrt {a b}, -4 \sqrt {2}\, \sqrt {a b}+1, \frac {\sqrt {2}\, \sqrt {a b}\, x -a}{\sqrt {2}\, \sqrt {a b}\, x +a}\right ) c_2 +{\mathrm e}^{\frac {-b \,x^{2}+a}{x}} \operatorname {HeunD}\left (4 \sqrt {2}\, \sqrt {a b}, -1-4 \sqrt {2}\, \sqrt {a b}, 8 \sqrt {2}\, \sqrt {a b}, -4 \sqrt {2}\, \sqrt {a b}+1, \frac {\sqrt {2}\, \sqrt {a b}\, x -a}{\sqrt {2}\, \sqrt {a b}\, x +a}\right ) c_1 \right ) \]

✓ Mathematica. Time used: 0.295 (sec). Leaf size: 38

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

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

✗ Sympy

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

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