55.29.45 problem 105

Internal problem ID [13878]
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 : 105
Date solved : Thursday, October 02, 2025 at 08:08:16 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

✓ Maple. Time used: 0.098 (sec). Leaf size: 73

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

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

✓ Mathematica. Time used: 0.234 (sec). Leaf size: 90

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

\begin{align*} y(x)&\to e^{-b (a+x)} (a+x)^{1-c} (-b (a+x))^{-c} \left (c_1 e^{a b} (a+x)^{a b} (-b (a+x))^c+b c_2 (-b (a+x))^{a b} (a+x)^c \Gamma (-a b+c-1,-b (a+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(b*y(x) + (a + x)*Derivative(y(x), (x, 2)) + (b*x + c)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False