61.29.31 problem 140

Internal problem ID [12561]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-4
Problem number : 140
Date solved : Wednesday, March 05, 2025 at 07:10:05 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+\left (a \,x^{2}+\left (a b -1\right ) x +b \right ) y^{\prime }+a^{2} b x y&=0 \end{align*}

✓ Maple. Time used: 1.155 (sec). Leaf size: 199

ode:=x^2*diff(diff(y(x),x),x)+(a*x^2+(a*b-1)*x+b)*diff(y(x),x)+a^2*b*x*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left ({\mathrm e}^{\frac {-a \,x^{2}+b}{x}} \operatorname {HeunD}\left (4 \sqrt {a b}, -b^{2} a^{2}+4 a b -8 \sqrt {a b}-4, -8 \sqrt {a b}\, \left (a b -1\right ), b^{2} a^{2}-4 a b -8 \sqrt {a b}+4, \frac {x \sqrt {a b}-b}{x \sqrt {a b}+b}\right ) c_{1} +\operatorname {HeunD}\left (-4 \sqrt {a b}, -b^{2} a^{2}+4 a b -8 \sqrt {a b}-4, -8 \sqrt {a b}\, \left (a b -1\right ), b^{2} a^{2}-4 a b -8 \sqrt {a b}+4, \frac {x \sqrt {a b}-b}{x \sqrt {a b}+b}\right ) c_{2} \right ) x^{1-\frac {a b}{2}} \]

✓ Mathematica. Time used: 0.782 (sec). Leaf size: 130

ode=x^2*D[y[x],{x,2}]+(a*x^2+(a*b-1)*x+b)*D[y[x],x]+a^2*b*x*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(a**2*b*x*y(x) + x**2*Derivative(y(x), (x, 2)) + (a*x**2 + b + x*(a*b - 1))*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -x*(-a**2*b*y(x) - x*Derivative(y(x), (x, 2)))/(a*b*x + a*x**2 + b - x) + Derivative(y(x), x) cannot be solved by the factorable group method