55.29.11 problem 71

Internal problem ID [13844]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-3
Problem number : 71
Date solved : Friday, October 03, 2025 at 06:55:06 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 40

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

\[ y = c_1 \,{\mathrm e}^{-x c}+c_2 \,x^{-\frac {b}{2}} \operatorname {WhittakerM}\left (-\frac {b}{2}, -\frac {b}{2}+\frac {1}{2}, \left (-2 c +a \right ) x \right ) {\mathrm e}^{-\frac {a x}{2}} \]

✓ Mathematica. Time used: 0.213 (sec). Leaf size: 50

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

\begin{align*} y(x)&\to e^{-c x} \left (c_1-c_2 x^{1-b} (x (a-2 c))^{b-1} \Gamma (1-b,(a-2 c) x)\right ) \end{align*}

✗ Sympy

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

ValueError : Expected Expr or iterable but got None