61.31.25 problem 206

Internal problem ID [12627]
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 : 206
Date solved : Monday, March 31, 2025 at 06:12:36 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.106 (sec). Leaf size: 65

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

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

✗ Mathematica

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

Timed out

✗ Sympy

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

False