61.29.8 problem 117

Internal problem ID [12538]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-4
Problem number : 117
Date solved : Monday, March 31, 2025 at 05:38:20 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-\left (a^{2} x^{4}+a \left (2 b -1\right ) x^{2}+b \left (b +1\right )\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 38

ode:=x^2*diff(diff(y(x),x),x)-(a^2*x^4+a*(2*b-1)*x^2+b*(1+b))*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.259 (sec). Leaf size: 70

ode=x^2*D[y[x],{x,2}]-(a^2*x^4+a*(2*b-1)*x^2+b*(b+1))*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{2} x^{-b} e^{-\frac {a x^2}{2}-b} \left (a c_2 x^{2 b+3} \left (-a x^2\right )^{-b-\frac {3}{2}} \Gamma \left (b+\frac {1}{2},-a x^2\right )+2 c_1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - (a**2*x**4 + a*x**2*(2*b - 1) + b*(b + 1))*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None