55.29.47 problem 107

Internal problem ID [13880]
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 : 107
Date solved : Friday, October 03, 2025 at 06:55:20 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.107 (sec). Leaf size: 166

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

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

✓ Mathematica. Time used: 0.985 (sec). Leaf size: 122

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

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

✗ Sympy

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

False