55.28.31 problem 41

Internal problem ID [13814]
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 : 41
Date solved : Thursday, October 02, 2025 at 08:07:07 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.035 (sec). Leaf size: 70

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

\[ y = \frac {{\mathrm e}^{-b x} \left ({\mathrm e}^{-\frac {a \,x^{4}}{8}} \operatorname {WhittakerM}\left (\frac {3}{8}, \frac {7}{8}, \frac {a \,x^{4}}{4}\right ) c_2 \,a^{2} x^{8}+\frac {7 \,{\mathrm e}^{-\frac {a \,x^{4}}{4}} 2^{{1}/{4}} c_2 \left (a \,x^{4}+3\right ) \left (a \,x^{4}\right )^{{11}/{8}}}{8}+c_1 \,x^{{13}/{2}}\right )}{x^{{11}/{2}}} \]

✓ Mathematica. Time used: 0.211 (sec). Leaf size: 51

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

\begin{align*} y(x)&\to \frac {1}{8} e^{-b x} \left (8 c_1 x-\sqrt {2} c_2 \sqrt [4]{a x^4} \Gamma \left (-\frac {1}{4},\frac {a x^4}{4}\right )\right ) \end{align*}

✗ Sympy

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

False