61.31.26 problem 207

Internal problem ID [12628]
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 : 207
Date solved : Thursday, March 13, 2025 at 11:57:14 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 \left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+3 \left (3 a \,x^{2}+2 b x +c \right ) y^{\prime }+\left (6 a x +2 b +\lambda \right ) y&=0 \end{align*}

✓ Maple. Time used: 1.701 (sec). Leaf size: 101

ode:=2*(a*x^3+b*x^2+c*x+d)*diff(diff(y(x),x),x)+3*(3*a*x^2+2*b*x+c)*diff(y(x),x)+(6*a*x+2*b+lambda)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 96.652 (sec). Leaf size: 3202

ode=2*(a*x^3+b*x^2+c*x+d)*D[y[x],{x,2}]+3*(3*a*x^2+2*b*x+c)*D[y[x],x]+(6*a*x+2*b+\[Lambda])*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Too large to display

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
d = symbols("d") 
cg = symbols("cg") 
y = Function("y") 
ode = Eq((6*a*x + 2*b + cg)*y(x) + (9*a*x**2 + 6*b*x + 3*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