55.29.26 problem 86

Internal problem ID [13859]
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 : 86
Date solved : Thursday, October 02, 2025 at 08:07:55 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+\left (a \,x^{3}+b \right ) y^{\prime }+a \left (b -1\right ) x^{2} y&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 99

ode:=x*diff(diff(y(x),x),x)+(a*x^3+b)*diff(y(x),x)+a*(b-1)*x^2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_2 \,x^{-4-\frac {b}{2}} {\mathrm e}^{-\frac {a \,x^{3}}{6}} \operatorname {WhittakerM}\left (\frac {1}{3}+\frac {b}{6}, \frac {b}{6}+\frac {5}{6}, \frac {a \,x^{3}}{3}\right ) a^{2} x^{6}+\frac {c_2 \,x^{-4-\frac {b}{2}} 3^{\frac {2}{3}-\frac {b}{6}} {\mathrm e}^{-\frac {a \,x^{3}}{3}} \left (b +5\right ) \left (a \,x^{3}+b +2\right ) \left (a \,x^{3}\right )^{\frac {4}{3}+\frac {b}{6}}}{9}+c_1 \,x^{-b +1} \]

✓ Mathematica. Time used: 0.187 (sec). Leaf size: 60

ode=x*D[y[x],{x,2}]+(a*x^3+b)*D[y[x],x]+a*(b-1)*x^2*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 x^{1-b}-3^{\frac {b-4}{3}} c_2 \left (a x^3\right )^{\frac {1}{3}-\frac {b}{3}} \Gamma \left (\frac {b-1}{3},\frac {a x^3}{3}\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(a*x**2*(b - 1)*y(x) + x*Derivative(y(x), (x, 2)) + (a*x**3 + b)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None