55.29.19 problem 79

Internal problem ID [13852]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-3
Problem number : 79
Date solved : Friday, October 03, 2025 at 06:55:12 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+\left (a \,x^{2} b +b -5\right ) y^{\prime }+2 a^{2} \left (b -2\right ) x^{3} y&=0 \end{align*}

✓ Maple. Time used: 0.125 (sec). Leaf size: 88

ode:=x*diff(diff(y(x),x),x)+(a*b*x^2+b-5)*diff(y(x),x)+2*a^2*(b-2)*x^3*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{-\frac {a \,x^{2} \left (b -2\right )}{2}} c_2 \left (-3 b \operatorname {KummerU}\left (\frac {b}{2}+1, -2+\frac {b}{2}, \frac {a \left (b -4\right ) x^{2}}{2}\right )+\left (a \left (b -4\right ) x^{2}+b +4\right ) \operatorname {KummerU}\left (\frac {b}{2}, -2+\frac {b}{2}, \frac {a \left (b -4\right ) x^{2}}{2}\right )\right )}{2}+{\mathrm e}^{-a \,x^{2}} c_1 \left (a \,x^{2}+1\right ) \]

✓ Mathematica. Time used: 2.126 (sec). Leaf size: 67

ode=x*D[y[x],{x,2}]+(a*b*x^2+b-5)*D[y[x],x]+2*a^2*(b-2)*x^3*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-a x^2} \left (a x^2+1\right ) \left (c_2 \int _1^x\frac {e^{-\frac {1}{2} a (b-4) K[1]^2} K[1]^{5-b}}{\left (a K[1]^2+1\right )^2}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(2*a**2*x**3*(b - 2)*y(x) + x*Derivative(y(x), (x, 2)) + (a*b*x**2 + b - 5)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None