55.28.13 problem 23

Internal problem ID [13796]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-2
Problem number : 23
Date solved : Friday, October 03, 2025 at 06:54:50 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.063 (sec). Leaf size: 75

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

\[ y = {\mathrm e}^{-\frac {\left (i \sqrt {b}\, x +\frac {3 a}{2}\right ) x^{2}}{3}} x \left (\operatorname {KummerM}\left (\frac {i c +4 \sqrt {b}}{6 \sqrt {b}}, \frac {4}{3}, \frac {2 i \sqrt {b}\, x^{3}}{3}\right ) c_1 +\operatorname {KummerU}\left (\frac {i c +4 \sqrt {b}}{6 \sqrt {b}}, \frac {4}{3}, \frac {2 i \sqrt {b}\, x^{3}}{3}\right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.119 (sec). Leaf size: 121

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

\begin{align*} y(x)&\to \frac {\sqrt [3]{2} \sqrt [3]{x^3} e^{\frac {1}{6} i x^2 \left (2 \sqrt {b} x+3 i a\right )} \left (c_1 \operatorname {HypergeometricU}\left (\frac {1}{3}-\frac {i c}{6 \sqrt {b}},\frac {2}{3},-\frac {2}{3} i \sqrt {b} x^3\right )+c_2 L_{\frac {i c}{6 \sqrt {b}}-\frac {1}{3}}^{-\frac {1}{3}}\left (-\frac {2}{3} i \sqrt {b} x^3\right )\right )}{x} \end{align*}

✗ Sympy

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

False