61.30.22 problem 170

Internal problem ID [12591]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-5
Problem number : 170
Date solved : Monday, March 31, 2025 at 05:52:32 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.318 (sec). Leaf size: 1260

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

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

✓ Mathematica. Time used: 2.533 (sec). Leaf size: 96

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

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

✗ Sympy

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

False