55.29.46 problem 106

Internal problem ID [13879]
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 : 106
Date solved : Friday, October 03, 2025 at 06:55:19 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (a_{1} x +a_{0} \right ) y^{\prime \prime }+\left (b_{1} x +b_{0} \right ) y^{\prime }-m b_{1} y&=0 \end{align*}

✓ Maple. Time used: 0.039 (sec). Leaf size: 101

ode:=(a__1*x+a__0)*diff(diff(y(x),x),x)+(b__1*x+b__0)*diff(y(x),x)-m*b__1*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {b_{1} x}{a_{1}}} \left (a_{1} x +a_{0} \right )^{\frac {a_{0} b_{1} +a_{1}^{2}-a_{1} b_{0}}{a_{1}^{2}}} \left (\operatorname {KummerU}\left (m +1, \frac {a_{0} b_{1} +2 a_{1}^{2}-a_{1} b_{0}}{a_{1}^{2}}, \frac {b_{1} \left (a_{1} x +a_{0} \right )}{a_{1}^{2}}\right ) c_2 +\operatorname {KummerM}\left (m +1, \frac {a_{0} b_{1} +2 a_{1}^{2}-a_{1} b_{0}}{a_{1}^{2}}, \frac {b_{1} \left (a_{1} x +a_{0} \right )}{a_{1}^{2}}\right ) c_1 \right ) \]

✓ Mathematica. Time used: 0.113 (sec). Leaf size: 102

ode=(a1*x+a0)*D[y[x],{x,2}]+(b1*x+b0)*D[y[x],x]-m*b1*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-\frac {\text {b1} x}{\text {a1}}} (\text {a0}+\text {a1} x)^{\frac {\text {a0} \text {b1}+\text {a1}^2-\text {a1} \text {b0}}{\text {a1}^2}} \left (c_1 \operatorname {HypergeometricU}\left (m+1,-\frac {\text {b0}}{\text {a1}}+\frac {\text {a0} \text {b1}}{\text {a1}^2}+2,\frac {\text {b1} (\text {a0}+\text {a1} x)}{\text {a1}^2}\right )+c_2 L_{-m-1}^{\frac {\text {a1}^2-\text {b0} \text {a1}+\text {a0} \text {b1}}{\text {a1}^2}}\left (\frac {\text {b1} (\text {a0}+\text {a1} x)}{\text {a1}^2}\right )\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a__0 = symbols("a__0") 
a__1 = symbols("a__1") 
b__0 = symbols("b__0") 
b__1 = symbols("b__1") 
m = symbols("m") 
y = Function("y") 
ode = Eq(-b__1*m*y(x) + (a__0 + a__1*x)*Derivative(y(x), (x, 2)) + (b__0 + b__1*x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False