55.29.30 problem 90

Internal problem ID [13863]
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 : 90
Date solved : Thursday, October 02, 2025 at 08:07:59 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.117 (sec). Leaf size: 42

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

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

✓ Mathematica. Time used: 0.955 (sec). Leaf size: 57

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

\begin{align*} y(x)&\to x^{1-d} \left (c_2 \int _1^x\exp \left (-\frac {1}{6} K[1] (6 c+K[1] (3 b+2 a K[1]))\right ) K[1]^{d-2}dK[1]+c_1\right ) \end{align*}

✗ Sympy

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

ValueError : Expected Expr or iterable but got None