23.3.215 problem 217

Internal problem ID [5929]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 217
Date solved : Friday, October 03, 2025 at 01:45:27 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.125 (sec). Leaf size: 106

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

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

✓ Mathematica. Time used: 0.057 (sec). Leaf size: 117

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

\begin{align*} y(x)&\to x^{2 a+1} e^{b x-\sqrt {b-1} \sqrt {b} x} \left (c_1 \operatorname {HypergeometricU}\left (\sqrt {\frac {b-1}{b}} a+a+1,2 a+2,2 \sqrt {b-1} \sqrt {b} x\right )+c_2 L_{-a \left (\sqrt {\frac {b-1}{b}}+1\right )-1}^{2 a+1}\left (2 \sqrt {b-1} \sqrt {b} x\right )\right ) \end{align*}

✗ Sympy

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

ValueError : Expected Expr or iterable but got None