61.30.21 problem 169

Internal problem ID [12590]
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 : 169
Date solved : Monday, March 31, 2025 at 05:52:28 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.213 (sec). Leaf size: 898

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

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 1.368 (sec). Leaf size: 64

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

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

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

False