61.31.22 problem 203

Internal problem ID [12624]
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 : 203
Date solved : Monday, March 31, 2025 at 06:09:29 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✗ Maple

ode:=x*(a*x^2+b*x+1)*diff(diff(y(x),x),x)+(alpha*x^2+beta*x+gamma)*diff(y(x),x)+(n*x+m)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 55.102 (sec). Leaf size: 524

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

\[ y(x)\to -\frac {2^{-\gamma } \left (-\frac {a x}{\sqrt {b^2-4 a}+b}\right )^{1-\gamma } \left (a \left (-2^{\gamma }\right ) c_1 x \left (-\frac {a x}{\sqrt {b^2-4 a}+b}\right )^{\gamma -1} \text {HeunG}\left [\frac {b-\sqrt {b^2-4 a}}{\sqrt {b^2-4 a}+b},\frac {2 m}{\sqrt {b^2-4 a}+b},\frac {1}{2} \left (-\sqrt {\frac {a^2+\alpha ^2-2 a (\alpha +2 n)}{a^2}}+\frac {\alpha }{a}-1\right ),\frac {1}{2} \left (\sqrt {\frac {a^2+\alpha ^2-2 a (\alpha +2 n)}{a^2}}+\frac {\alpha }{a}-1\right ),\gamma ,\frac {2 a^2 \gamma -a \left (\beta \left (\sqrt {b^2-4 a}+b\right )+2 \alpha \right )+\alpha b \left (\sqrt {b^2-4 a}+b\right )}{a \left (b \left (\sqrt {b^2-4 a}+b\right )-4 a\right )},-\frac {2 a x}{\sqrt {b^2-4 a}+b}\right ]-2 a c_2 x \text {HeunG}\left [\frac {b-\sqrt {b^2-4 a}}{\sqrt {b^2-4 a}+b},\frac {2 ((\gamma -1) (b \gamma -\beta )+m)}{\sqrt {b^2-4 a}+b},\frac {1}{2} \left (-\sqrt {\frac {a^2+\alpha ^2-2 a (\alpha +2 n)}{a^2}}+\frac {\alpha }{a}-2 \gamma +1\right ),\frac {1}{2} \left (\sqrt {\frac {a^2+\alpha ^2-2 a (\alpha +2 n)}{a^2}}+\frac {\alpha }{a}-2 \gamma +1\right ),2-\gamma ,\frac {2 a^2 \gamma -a \left (\beta \left (\sqrt {b^2-4 a}+b\right )+2 \alpha \right )+\alpha b \left (\sqrt {b^2-4 a}+b\right )}{a \left (b \left (\sqrt {b^2-4 a}+b\right )-4 a\right )},-\frac {2 a x}{\sqrt {b^2-4 a}+b}\right ]\right )}{a x} \]

✗ Sympy

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

ValueError : Expected Expr or iterable but got None