61.31.28 problem 209

Internal problem ID [12630]
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 : 209
Date solved : Thursday, March 13, 2025 at 11:59:12 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 12.691 (sec). Leaf size: 76

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

\[ y = \left (x +\lambda \right ) \left (\left (\int {\mathrm e}^{-\int \frac {x^{4}+\left (2 a +\lambda \right ) x^{3}+2 b \,x^{2}+\left (\lambda ^{3}+2 c \right ) x +\lambda ^{4}+2 d}{\left (a \,x^{3}+b \,x^{2}+c x +d \right ) \left (x +\lambda \right )}d x}d x \right ) c_{2} +c_{1} \right ) \]

✓ Mathematica. Time used: 1.7 (sec). Leaf size: 103

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

\[ y(x)\to \frac {(\lambda +x) \left (c_2 \int _1^x\exp \left (\int _1^{K[2]}-\frac {\lambda ^4+K[1] \lambda ^3+K[1]^3 \lambda +K[1]^4+2 a K[1]^3+2 b K[1]^2+2 d+2 c K[1]}{(\lambda +K[1]) (d+K[1] (c+K[1] (b+a K[1])))}dK[1]\right )dK[2]+c_1\right )}{\lambda } \]

✗ 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((cg**3 + x**3)*Derivative(y(x), x) - (cg**2 - cg*x + x**2)*y(x) + (a*x**3 + b*x**2 + c*x + d)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False