55.32.16 problem 197

Internal problem ID [13970]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-6
Problem number : 197
Date solved : Friday, October 03, 2025 at 07:04:15 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 1.054 (sec). Leaf size: 2132

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

\[ \text {Expression too large to display} \]

✓ Mathematica. Time used: 5.071 (sec). Leaf size: 186

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

\begin{align*} y(x)&\to -\frac {\left (-k x (a x+2)-(b-1) x^2+c (k-2 x)+k^2\right ) \left (c_2 \int _1^x\frac {\exp \left (\frac {(2 c+b k) \arctan \left (\frac {b+2 a K[1]}{\sqrt {4 a c-b^2}}\right )}{c \sqrt {4 a c-b^2}}\right ) K[1]^{-\frac {k}{c}} (c+K[1] (b+a K[1]))^{\frac {k}{2 c}+1}}{\left (k^2-K[1] (a K[1]+2) k-(b-1) K[1]^2+c (k-2 K[1])\right )^2}dK[1]+c_1\right )}{a k+b-c (k-2)-k^2+2 k-1} \end{align*}

✗ Sympy

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

ValueError : Expected Expr or iterable but got None