61.31.24 problem 205

Internal problem ID [12626]
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 : 205
Date solved : Thursday, March 13, 2025 at 11:53:53 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 ^{2}+x^{2}\right ) y^{\prime }+\left (x +\lambda \right ) y&=0 \end{align*}

✓ Maple. Time used: 1.524 (sec). Leaf size: 84

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

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

✗ Mathematica

ode=(a*x^3+b*x^2+c*x+d)*D[y[x],{x,2}]-(x^2-\[Lambda]^2)*D[y[x],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") 
cg = symbols("cg") 
y = Function("y") 
ode = Eq((cg + x)*y(x) - (-cg**2 + x**2)*Derivative(y(x), 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